\documentclass[11pt,notitlepage]{article}
\usepackage{amsmath, amssymb}
\usepackage{graphicx, stmaryrd, datetime, amscd, multicol, txfonts, ../picins, mathabx, dbnsymb, tensor, needspace, mathtools}
\usepackage[normalem]{ulem} % For \sout.
%\usepackage{mathbbol} % Needed for \bbe.
\usepackage{bbm} % Needed for \bbe.
\usepackage[margin=0.25in,foot=0.2in]{geometry}
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}
\usepackage{pdfcomment}
\usepackage[all]{xy}
\usepackage{tikz}
\usepackage{insdljs,rangen}
\usepackage[greek,english]{babel}
% Cyrillic letters following http://tex.stackexchange.com/questions/14633/what-packages-will-let-me-use-cyrillic-characters-in-math-mode:
% ... a better solution at pensieve://2015-11/xtx/xtx.tex, except if fails here because of "too many fonts".
\usepackage[T2A,T1]{fontenc}
\makeatletter
\def\easycyrsymbol#1{\mathord{\mathchoice
  {\mbox{\fontsize\tf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
  {\mbox{\fontsize\tf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
  {\mbox{\fontsize\sf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
  {\mbox{\fontsize\ssf@size\z@\usefont{T2A}{\rmdefault}{m}{n}#1}}
}}
\makeatother
\newcommand{\Zhe}{{\easycyrsymbol{\CYRZH}}}
\newcommand{\zhe}{{\easycyrsymbol{\cyrzh}}}

% Modified from http://tex.stackexchange.com/questions/37297/how-to-get-element-position-in-latex:
\usepackage{zref-user, zref-abspos}
\newwrite\posfile
\openout\posfile=\jobname.pos\relax
\makeatletter
\newcommand\dimtoin[1]{%
    \strip@pt\dimexpr 0.013837\dimexpr#1\relax\relax%
}
\makeatother

% From http://tex.stackexchange.com/questions/1429/latex-package-to-generate-qr-codes:
%\usepackage{pst-barcode}
%\usepackage{auto-pst-pdf}
% From http://tex.stackexchange.com/questions/89649/pgf-tikz-qr-code-generator
\usepackage[]{qrcode}

\usepackage[inline]{enumitem}
\setlist{nosep}
\setlist[itemize]{left=0pt .. 10pt}
\setlist[enumerate]{left=0pt .. 12pt}

\usepackage{twemojis} % For \twemoji{notebook}

% \llangle and \rrangle from https://www.latex4technics.com/?note=2dg7:
\makeatletter
\newsavebox{\@brx}
\newcommand{\llangle}[1][]{\savebox{\@brx}{\(\m@th{#1\langle}\)}%
  \mathopen{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}
\newcommand{\rrangle}[1][]{\savebox{\@brx}{\(\m@th{#1\rangle}\)}%
  \mathclose{\copy\@brx\kern-0.5\wd\@brx\usebox{\@brx}}}
\makeatother

\def\sheeturl{{\url{http://drorbn.net/AcademicPensieve/Projects/Agenda/}}}
\def\myurl{http://www.math.toronto.edu/~drorbn}
%\def\arXiv#1{{\qrcode[height=1em,level=L,nolink]{arxiv.org/abs/#1}\,\href{http://front.math.ucdavis.edu/#1}{arXiv:\linebreak[0]#1}}}
\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{\small #1}}}}
\def\bbs#1#2#3{{\href{http://drorbn.net/bbs/show?shot=#1-#2-#3.jpg}{BBS:\linebreak[0]#1-\linebreak[0]#2}}}
\def\mo#1{{\href{https://mathoverflow.net/questions/#1}{{\footnotesize m{\bf\sl o}:}\linebreak[0]{\small #1}}}}

\write\posfile{Entries=List[}
\newcounter{entry}
\newcommand\entry[1]{\phantomsection\label{entry:#1}%
\stepcounter{entry}%
{\red\bf\tiny\strut\zsavepos{#1}\write\posfile{  Entry[\thepage, "#1", \dimtoin{\zposx{#1}sp}, \dimtoin{\zposy{#1}sp}],}%
#1}}
\newcommand\archived[1]{{\phantomsection\label{entry:#1}\bf\gray\tiny #1}}
\newcommand\refentry[1]{\hyperref[entry:#1]{{\small$\to$p\pageref*{entry:#1}:\bf #1}}}

%\paperwidth 8in \paperheight 10.5in
%\textwidth 8in \textheight 10.5in
%\oddsidemargin -0.75in \evensidemargin \oddsidemargin \topmargin -0.75in
%\headheight 0in \headsep 0in \footskip 0in
\parindent 0in
%\setlength{\topsep}{0pt}
%\pagestyle{empty}
\dmyydate

% Following http://tex.stackexchange.com/questions/23521/tabular-vertical-alignment-to-top:
\def\imagetop#1{\vtop{\null\hbox{#1}}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\Ad{\operatorname{Ad}}
\def\ad{\operatorname{ad}}
\def\adj{\operatorname{adj}}
\def\ann{{\operatorname{ann}}}
\def\Arg{\operatorname{Arg}}
\def\aug{\operatorname{aug}}
\def\Aut{\operatorname{Aut}}
\def\attr{\operatorname{\mathfrak{tr}}}
\def\atdiv{\operatorname{div}}
\def\coker{\operatorname{coker}}
\def\crit{\operatorname{crit}}
\def\der{\operatorname{der}}
\def\Emb{\operatorname{Emb}}
\def\End{\operatorname{End}}
\def\Fun{\operatorname{Fun}}
\def\gr{\operatorname{gr}}
\def\GRT{\operatorname{GRT}}
\def\hgt{\operatorname{ht}}
\def\Hom{\operatorname{Hom}}
\def\im{\operatorname{im}}
\def\Imm{\operatorname{Imm}}
\def\Inn{\operatorname{Inn}}
\def\lk{\operatorname{lk}}
\def\mal{\operatorname{mal}}
\def\Mal{\operatorname{Mal}}
\def\Mat{\operatorname{Mat}}
\def\Mod{\operatorname{Mod}}
\def\mor{\operatorname{mor}}
\def\obj{\operatorname{obj}}
\def\Out{\operatorname{Out}}
\def\rad{\operatorname{rad}}
\def\Rker{\operatorname{Rker}}
\def\sign{\operatorname{sign}}
\def\surjects{{\twoheadrightarrow}}
\def\wt{\operatorname{wt}}

\def\qed{{\hfill\text{$\Box$}}}
\def\remove{\!\setminus\!}
\def\FA{{F\!A}}
\def\FG{{F\!G}}
\def\FL{{F\!L}}
\def\St{S\!t}
\def\SL{{S\!L}}
\def\tr{\operatorname{tr}}

\def\sder{\operatorname{\mathfrak{sder}}}
\def\Span{\operatorname{span}}
\def\tder{\operatorname{\mathfrak{tder}}}
\def\uB{\text{\it u\!B}}
\def\uT{{u\calT}}
\def\vB{\text{\it v\!B}}
\def\vK{{v\calK}}
\def\vT{{v\calT}}
\def\wB{\text{\it w\!B}}
\def\wK{{w\calK}}
\def\wT{{w\calT}}
\def\PB{\text{\it P\!B}}
\def\PuB{\text{\it P\!u\!B}}
\def\PaB{\text{\it PaB}}
\def\PvB{\text{\it P\!v\!B}}
\def\PvOUB{\text{\it P\!v\!O\!U\!B}}
\def\PvT{\text{\it P\!v\!T}}
\def\PwB{\text{\it P\!w\!B}}

\def\barA{{\bar{A}}}
\def\barT{{\bar{T}}}
\def\bb1{{\mathbbm{1}}}
\def\bbA{{\mathbb A}}
\def\bbB{{\mathbb B}}
\def\bbC{{\mathbb C}}
\def\bbD{{\mathbb D}}
\def\bbe{\mathbbm{e}}
\def\bbE{\mathbb{E}}
\def\bbH{{\mathbb H}}
\def\bbi{\mathbb{i}}
\def\bbK{{\mathbb K}}
\def\bbN{{\mathbb N}}
\def\bbO{{\mathbb O}}
\def\bbP{{\mathbb P}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\calA{{\mathcal A}}
\def\calB{{\mathcal B}}
\def\calC{{\mathcal C}}
\def\calD{{\mathcal D}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calI{{\mathcal I}}
\def\calK{{\mathcal K}}
\def\calL{{\mathcal L}}
\def\calM{{\mathcal M}}
\def\calO{{\mathcal O}}
\def\calP{{\mathcal P}}
\def\calR{{\mathcal R}}
\def\calS{{\mathcal S}}
\def\calT{{\mathcal T}}
\def\calU{{\mathcal U}}
\def\calV{{\mathcal V}}
\def\calW{{\mathcal W}}
\def\calZ{{\mathcal Z}}
\def\fraka{{\mathfrak a}}
\def\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakh{{\mathfrak h}}
\def\frakl{{\mathfrak l}}
\def\frakn{{\mathfrak n}}
\def\frakt{{\mathfrak t}}

\def\krv{\mathfrak{krv}}
\def\RP{\bbR{\mathrm P}}

\def\eps{\epsilon}

\def\gray{\color{gray}}
\def\green{\color{green}}
\def\magenta{\color{magenta}}
\def\red{\color{red}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\yellowt#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{#1}}}

\begin{document}
%\setlength{\jot}{0ex}
\setlength{\abovedisplayskip}{0.5ex}
\setlength{\belowdisplayskip}{0.5ex}
\setlength{\abovedisplayshortskip}{0ex}
\setlength{\belowdisplayshortskip}{0ex}

\parbox[b]{4.3in}{
  {\LARGE\bf Dror Bar-Natan: Mathematical Monoblog}
  }
\hfill\parbox[b]{3.4in}{\tiny
  \null\hfill\sheeturl
  \newline\null\hfill initiated October 7, 2013; modified \today, \ampmtime
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/Ag}
}

\vskip -3mm
\rule{\textwidth}{1pt}
\vspace{-8mm}

\begin{multicols*}{2}

\entry{240628} {\bf Q.} Is $\mathfrak{grt}\simeq\mathfrak{krv}$ a version of ``braids act nearly transitively on simple curves''?

\entry{240624} {\bf Proj.} Study ``timelined tangles''. Are there homomorphic expansions?

\entry{240523} What if Conway preceded Alexander?

\entry{240516} \href{http://drorbn.net/AcademicPensieve/Projects/HigherRank/TheCast.pdf}{AP: Projects: HigherRank: TheCast} has the Feynman ring for $sl_3$, Heisenberg version.

\entry{240510} Started \href{https://drorbn.net/AP/People/Kuno/CheatSheet_Kuno.pdf}{AP: People: Kuno: CheatSheet}.

\vskip -2mm
%\needspace{0mm} % 0mm is not enough.
\parpic[r]{$\xymatrix@C=8mm@R=5mm{
  \{\PvOUB_{X,Y}\} \ar[r]^<>(0.5){Z\coloneqq X\sqcup Y} \ar[d] & \{\PvB_Z\} \ar[d] \\
  \{A^{\otimes X}\otimes B^{\otimes Y}\} \ar[r]^<>(0.5){Z\coloneqq X\sqcup Y} & \{D^{\otimes Z}\}
}$}
\entry{240502} The Drinfel'd Double: Given bialgebras $A$, $B$, an invertible non-degenerate $R\in A\otimes B$ defines double-meta-bialgebra morphism $\{\PvOUB_{X,Y}\} \to \{A^{\otimes X}\otimes B^{\otimes Y}\}$.
{\bf Thm?} Up to ??, there is a unique bialgebra structure on $D\coloneqq A\otimes B$ such that the diagram on the right commutes.

\entry{240322} \href{http://drorbn.net/AcademicPensieve/Projects/HigherRank}{AP: Projects: HigherRank}: Rank-2 Gassner:
\par\includegraphics[width=\linewidth]{figs/R2G.pdf}
The last line is a ``cross-rep glow''. {\bf Prob.} Find all cross-rep glows between products of Gassners at different $T$ parameters. There too: a more general rank 2 Burau.

\entry{240415} What are burves, plane curves mod braid moves?

\entry{240404} MAT 198 Cryptology dreams: Enigma, compression, DES, RSA, Bible codes, Bitcoin, homomorphic computation.

\entry{240401} \href{http://drorbn.net/AcademicPensieve/Projects/APAI/nb/APAILinks2.pdf}{Projects: APAI: APAILinks2.nb}: Why does $\rho_1$ makes sense for links? Does it always vanish when the MVA does? An MV$\rho_1$?

\entry{240328} {\bf Proj.} Say ``Rasmussen's $s$'' w/o saying ``spectral sequence''.

\entry{240305a} Boninger's conjecture: The coefficients of $\rho_1(K)$ in $z$ are uniform-sign for positive $K$. Tested in {\tt Rho\_d-Positivity.nb} in \href{http://drorbn.net/AcademicPensieve/Talks/Oaxaca-2210}{AP: T-: Oaxaca-2210}. True for $\Delta$ by Cromwell's ``Homogeneous Links''.

\entry{240319} Hartley: {\em The Conway Potential Function for Links}. Benheddi, Cimasoni: {\em Link Floer Homology Categorifies the Conway Function}.

\entry{240311} \href{http://drorbn.net/AcademicPensieve/2024-03}{2024-03}: Implementation of Kauffman States for tangles. Is there a pull-push formalism for PA operations?

\entry{231114} Ozsv\`ath, Szab\`o: \arXiv{1603.06559}, {\em Kauffman states, bordered algebras, and a bigraded knot invariant}, \arXiv{2212.11885}, {\em The pong algebra}, \arXiv{2311.07503}, {\em Planar graphs deformations of bordered knot algebras}. Zibrowius: \arXiv{1601.04915}, {\em Kauffman states and Heegaard diagrams for tangles}. Roberts: \arXiv{math/0607244}, \href{https://msp.org/agt/2009/9-1/p03.xhtml}{AGT 2009}, {\em Heegaard-Floer homology and string links}.

\entry{240305b} \twemoji{notebook}: Kauffman states $\leftrightarrow$ tree-dual-tree pairs in the checkerboard graphs $\leftrightarrow$ one-long-path smoothings. 

\entry{240304} Is the a local / size respecting description of $\text{knots}\hookrightarrow\text{bnots}$? In some completion? Being liberal about $R1$s? Allowing virtuals?

\entry{240302} {\bf JFF.} Compute the sign of a length $10^6$ permutation, 10 of whose entries are hidden.

\entry{240301} {\bf Q.} A local description of the Wirtinger-Alexander $\det$-signs?

\entry{240221b} In Kauffman's {\em Formal Knot Theory,} A state model for the {\em normalized} Alexander polynomial. Implementation: {\tt VxF\_Alexander.nb} in \href{http://drorbn.net/AcademicPensieve/People/Martchenkov}{AP: Pe-: Martchenkov}. \refentry{230609}.

\entry{240221a} ``The Alexander module dogma is wrong''.

\entry{170317} \href{https://en.wikipedia.org/wiki/Q-derivative}{Wikipedia: $q$-derivative}: $D_{q,x}f(x) = \frac{f(qx)-f(x)}{qx-x}$; has $D_{q,x}\bbe_q^x=\bbe_q^x$ (and $\bbe_q^0=1$); seek it and $\bbe_q^x$ and $xy=qyx$ in nature. Finds: $[a,x]=x$ $\Rightarrow$ $\bbe^{ta}x = \bbe^t x\bbe^{ta}$. Also, in tensor powers with $X_k\coloneqq \bbe^{t(a_1+\ldots+a_{k-1})}x_k$, have $X_kX_l=\bbe^tX_lX_k$ for $k<l$.

\entry{240213} {\bf Q.} Do braids act on multi-Fox profiles of free words?

\entry{190320} {\bf Proj.} Analyze \href{http://drorbn.net/AcademicPensieve/2016-06/nb/TurboGassner.pdf}{2016-06/Turbo-Gassner} (also \href{http://drorbn.net/AcademicPensieve/Talks/Toronto-1912/nb/GvIExamples.pdf}{Talks/Toronto-1912/GvIExamples}). Is it homological? Find it in Artin's representation. Following \href{http://drorbn.net/dbnvp/M19_Ito.php}{Ito@M19}, relations with Garside lengths?

\entry{240209b} Jim Davis: A Witt-valued quadratic pushforward story?

\entry{240209a} Calaque, Roca i Lucio \arXiv{2402.05539} {\em Associators from an operadic point of view}, a survey of associators.

\entry{240205} {\bf Q.} Is there a {\em (poly-)computable} functor from measured v.s.\ to Sets, extending Gaussian integration?

\entry{240130} Garoufalidis, Kashaev \arXiv{2311.11528} {\em Multivariable knot polynomials from braided Hopf algebras with automorphisms}. Also has a Drinfel'd double alternative.

\entry{240122} \twemoji{notebook}: With $Z$ a domain and $Q$ its field of fractions, given $\partial\colon R\to G$ invertible over $Q$ and $(\cdot,\cdot)\colon R\otimes G\to Z$ with $(r_1,\partial r_2) = (r_2,\partial r_1)$ get a symmetric $\langle\cdot,\cdot\rangle\colon(\coker_Z\partial)^{\otimes 2}\to Q/Z$ by $\langle\bar{g}_1,\bar{g}_2\rangle \coloneqq (\partial^{-1}_Qg_1,g_2)/Z$. {\bf Q.} When do two presentations yield equivalent $\langle\cdot,\cdot\rangle$'s?

\entry{240118} {\bf Proj.} Understand ``tangle nullities'' (cf.\ knot nullities).

\entry{240115} Borodzik, A.~Conway, Politarczyk \arXiv{2111.10632}, Section~1.1: TL signatures from matrices presenting Blanchfield. Kearton (1978): A edge-centric matrix presenting Blanchfield.

\entry{240103} Formal perturbed Gaussian integration, sans the determinant prefactor, is invariant under {\em all} linear coordinate changes.

\entry{240102b} \href{http://drorbn.net/AcademicPensieve/Projects/APAI/nb/PerturbedGaussianIntegration.pdf}{Projects: APAI: PerturbedGaussianIntegration.nb}: With $K$ a knot diagram, $w$ its writhe, $\pi K$ its rotation by $180^\circ$, $\calR_1$ the integrand for $\rho_1$,
$\calP = \left\{{x_{2n+1}\to p_1,}\atop{p_{2n+1}\to x_{2n+1}}\right\} \cup \bigcup\limits_{c:(s,i,j)\in K} \left\{{x_i\to T^{\nu_i}(p_1-p_{i+1}),\ p_i\to T^{-\nu_i}x_i,\ p_j\to T^{-\nu_i-s}x_j,} \atop {x_j\to T^{\nu_i}((1-T^s)p_{i+1}+T^sp_1-p_{j+1})}\right\}$ with $\nu$ the Alexander numbering,
get $\mathrlap{\int}{\scriptstyle G}\overline{\calR_1(\pi K)} = T^{-w}\mathrlap{\int}{\scriptstyle G}\calR_1(K)/\calP$.

\entry{240102a} \href{http://drorbn.net/AcademicPensieve/2023-12/nb/BridgesAndTunnels.pdf}{2023-12: BridgesAndTunnels.nb}: Palindromicity using bridges and tunnels.

\entry{231214b} {\bf Q.} Are there ``dual presentations'' signature formulas?

\entry{231214a} {\bf Q.} Do virtual knots have a meaningful Dehn fundamental group? Is it related to the two Wirtinger ones? Are there others?

\entry{231212} In the knot complement cyclic cover context, $0 \to H_1(\tilde{X}) \to H_1(\tilde{X},\tilde{p}) \underset{s}{\overset{\partial}{\rightleftarrows}} \ker(i_*) \to 0$ is split exact, where $\bbZ[T^{\pm 1}] = H_0(\tilde{p}) \overset{i_*}{\longrightarrow} H_0(\tilde{X}) = \bbZ$ so $H_1(\tilde{X}) \simeq \ker\partial \simeq H_1(\tilde{X},\tilde{p})/\im s$.

\vskip -2mm
\needspace{6mm} % 5mm is not enough.
\parpic[r]{$\xymatrix@C=7mm@R=5mm{
  R_1 \ar@<2pt>[r]^{\alpha_1} \ar@<2pt>[d]^{\gamma_1} &
  G_1 \ar@<2pt>[d]^{\beta_1} \ar@<2pt>[l]^{\eta_1} \\
  R_2 \ar@<2pt>[r]^{\alpha_2} \ar@<2pt>[u]^{\gamma_2} &
  G_2 \ar@<2pt>[u]^{\beta_2} \ar@<2pt>[l]^{\eta_2}
}$}
\picskip{4}
\entry{230811} {\bf Thm} (cf.\ Lickorish pp.\ 50). Module presentations $R_i \overset{\alpha_i}{\rightarrow}G_i\ (\rightarrow M,\ i=1,2$) are equivalent iff $\exists\,\beta_i,\gamma_i,\eta_i$ as here s.t.\ $\beta_1\alpha_1 = \alpha_2\gamma_1$, $\beta_2\alpha_2 = \alpha_1\gamma_2$, $\beta_2\beta_1+\alpha_1\eta_1 = I$, and $\beta_1\beta_2+\alpha_2\eta_2 = I$.
Then $\begin{pmatrix}I&0\\\beta_1&\alpha_2\end{pmatrix} = \begin{pmatrix}\alpha_1&\beta_2\\0&I\end{pmatrix} \begin{pmatrix}\eta_1&-\gamma_2\\\beta_1&\alpha_2\end{pmatrix}$ and $\begin{pmatrix}\alpha_1&\beta_2\\0&I\end{pmatrix} = \begin{pmatrix}I&0\\\beta_1&\alpha_2\end{pmatrix} \begin{pmatrix}\alpha_1&\beta_2\\-\gamma_1&\eta_2\end{pmatrix}$ so elementary ideals make sense and (if $\exists\alpha_i^{-1}$) $\alpha_1^{-1} = \gamma_2\alpha_2^{-1}\beta_1+\eta_1$ and $\alpha_2^{-1} = \gamma_1\alpha_1^{-1}\beta_2+\eta_2$.
Implement for all Alexander presentations!

\entry{230406} The AKKN operational envelope: Cuts, then doubles (and reversals), then letter substitutions, then infusion of constants, then merges.

\entry{231205} L\'opez Neumann, van der Veen: \arXiv{2312.02070}, ``Genus bounds from unrolled quantum groups at roots of unity''.

\needspace{7mm} % 6mm is not enough.
\vskip -2mm
\parpic[r]{\scalebox{1}{\input{figs/ReducedPentagon.pdf_t}}}
\entry{230404} {\bf Q.} Is the pentagon in emergent 2-poles 2-strands equivalent to the standard $\FL$-pentagon? (The spaces grow slower!) Can $\Phi_{pps}$ be found degree by degree? Is there a GT group? Isomorphic to a known variant?

\entry{220720} In framed $\Sigma^2$, mod homotopy, curves are the same as ``sailing curves'', immersions whose tangent avoids direct front winds.

\entry{231105a} Implicit in Chrisman, Todd \arXiv{2307.09387}: $\Zhe(D)=D\cup\omega$, where $\omega\in H^1(\underline{D})$ is the self-intersection 1-form of $\underline{D}$, the underlying graph of $D$. Is there a rotational version? \refentry{141113b}

\entry{231103} Tautology: Alexander numbering $\Leftrightarrow$ homologically trivial.

\entry{231013b} {\bf Def.} A Measured Partial Quadratic (MPQ) on a v.s.\ $V$ is a quadratic $Q$ defined on a subspace $D\subset V$ along with a volume element on $D$.

{\bf Conj.} Given $\phi\colon V\to W$ and an MPQ $Q$ on $V$ there is a unique MPQ $\phi_*Q$ on $W$ such that for every quadratic $U$ on $W$, $\det(U+\phi_*Q) = \det(Q+\phi^*U)$ (``\st{quadratic reciprocity}'').

\entry{210818} Abbasi's $u\calK\hookrightarrow v\calK$, ``opposite inner-most pairs of non-local R2s can be removed'': An ``even set of xings'' has even incidence with every face. Non-empty ones transport through local R-moves, get created when a non-local R2 is performed, and unions of even sets can be projected preserving R-moves. Now use Abbasi's zigzag trick to reduce the xing-number profile between non-local moves. {\bf Qs.} Links? Rotational virtuals?  Braid-like R-moves? Does $u\calK\hookrightarrow v\calK$ preserve 1-cycles? Can these ideas be used to prove Satoh's conjecture?

\needspace{7mm} % 6mm is not enough.
\vskip -2mm
\parpic[r]{\includegraphics[width=1.8in]{figs/ktgs_page_8.png}}
\entry{231014} \href{https://www.math.toronto.edu/~drorbn/papers/ktgs/}{Papers/ktgs}: Why the dots? $\bullet \to \nu^{1/2}$, with $\nu = Z(\bigcirc)$.

\entry{231013a} Turaev's \arXiv{math/0310218} ``Virtual Strings'' has ``based matrices'' and sliceness criteria.

\entry{231009} Best $w$-practices:
\begin{enumerate}
\item Trivalent tangles are end-labeled, to make a circuit algebra.
\item Tubes are bare (no colours and/or orientations).
\item Crossings have no signs; filtration is by comparison with virtual crossings.
\item Vertices are oriented and have marked legs: stem ($s$), upper ($u$), and lower ($l$). They are classical: they satisfy both R4s.
\item Wenjugating interchanges the two vertex types, and adds a virtual $l\leftrightarrow u$ crossing.
\item Only stems can be unzipped. Unzipping unwened edges connects $u$ to $u$ above a connection of $l$ to $l$. Unzipping through a wen is defined by wenjugating it out.
\end{enumerate}
\vskip 4pt
\[ \input{figs/wpractices.pdf_t} \]

\entry{231010} In \arXiv{1612.05641}, a factorization of half-Dehn:
\newline\includegraphics[width=\linewidth]{figs/ArXiv-1612.05641-FactorizationOfDehn.png}

\entry{231008} Is there a finitely presented algebraic structure made of unoriented pure tangles?

\entry{231002b} \href{http://drorbn.net/AcademicPensieve/Projects/FullDoPeGDO/nb/cR.pdf}{Projects: FullDoPeGDO: cR.nb}: in $sl_{2+}^0$,
$(x_1,\,x_2)\cdot R_{12} = R_{12}\cdot(x_1+(1-B_1)x_2,\,B_1x_2)$,
and
$(y_1,\,y_2)\cdot R_{12} = R_{12}\cdot(y_1,\,\frac{b_2(1-B_1^{-1})}{b_1}y_1+B_1^{-1}y_2)$.

\entry{231002a} {\bf Proj.} ``Seven formulas for $\rho_1$''.

\entry{221228} {\bf Missing.} A fully defined theory of pushing forward Gaussians (better with determinants and signatures). {\bf Q.} Does the signature pushforward work also for $\det'$?

\entry{230915} Frohlich's trace: Let $A=\calU(sl_{2+}^0)$ and $\phi\colon\bbQ[b,z,a] \to A_A$ by $b^kz^na^m\mapsto b^ky^na^mx^n/n!$ (surjective as $0 = [a,b^ky^na^mx^l] = (l-n)b^ky^na^mx^l$ in $A_A$).
In $A$ with $f=f(a)$ and $\nabla f\coloneqq f(a)-f(a-1)$, $[x,f]=-\nabla f\cdot x$ so in $A_A$, $0 = [x,y^{n+1}fx^n] = (n+1)by^nfx^n-y^{n+1}\nabla f x^{n+1}$ so $\phi(bz^nf) = \phi(z^{n+1}\nabla f)$.
Ergo $\calG(\tr)$: $\tr(\bbe^{\beta b}\bbe^{\eta y}\bbe^{\alpha a}\bbe^{\xi x}) = \phi(\bbe^{\beta b}\bbe^{\eta\xi z}\bbe^{\alpha a}) = \phi(\bbe^{\beta z\nabla}\bbe^{\eta\xi z}\bbe^{\alpha a}) = \phi(\bbe^{\alpha a+(\eta\xi+\beta(1-\bbe^{-\alpha}))z})$. \refentry{200906}, \refentry{180909a}.

\entry{230911} $\calA(*_*)$ is a contraction algebra.

\entry{230904} If cobrackets come from the asymmetry of coproducts, wherefore the Turaev cobracket?

\entry{230823} With $\Lambda\coloneqq\bbZ[T^{\pm 1}]$ and $\Lambda_0 \coloneqq \bbZ[T+T^{-1}]$, is it that for every $\Lambda$-module $A$ there is a $\Lambda_0$-module $B$ with $A\oplus\barA \equiv \Lambda\otimes_{\Lambda_0}B$?

\entry{230822} Figure out the Kashaev-formula weight system (for $\Delta^2$?).

\entry{230817} Is there a Goeritz formula for $\Delta$?

\entry{230228b} In \href{http://drorbn.net/AcademicPensieve/People/VanDerVeen/VanDerVeen_Journal.pdf}{VanDerVeen\_Journal}: Given a diagram $D$ for a long $K$, the phase $\phi$ along a curve $\gamma\subset D^c$ multiplies by $T^s$ whenever $\gamma$ passes over $D$ with sign $s$. {\bf Conj.} $\lk_K(\alpha,\beta) = (T-1)\langle$flow: generated by $\alpha$, measured by $\beta\rangle\,+\,\langle$phased $\alpha$ over $\beta$ count$\rangle$. $\alpha$ generates $\pm\phi$-flow when it runs over $D$. $\beta$ measures $\pm\phi^{-1}$-flow when it runs under $D$.

\entry{230804b} Fox derivatives: in $H_1(\tilde{D}_n,\tilde{p})$, $[\gamma] = \sum(\partial_i\gamma)[x_i]$.

\entry{230804a} Given a long $K$, is $K=(-K)$ mod $\Delta\Delta$?

\entry{140422} Yajima's ``On Simply Knotted Spheres in $\bbR^4$'': all are ribbon. {\em Pf.} Enough: every simply-knotted balloon forest is equivalent to a ribbon-certificate. Take an inner-most double line on a balloon, slide out string ends and string transverses, and compress to a new string. When no double lines are left, float balloons to un-nest them, and iron wens to make string ends external. See \bbs{KAL}{140520}{141129}. I don't fully understand the case of tubes.

\entry{230725} In w, what about $\frakg\ltimes(\frakg^*\oplus\frakg^*)$? And flying graphs?

\entry{230702} \href{http://drorbn.net/bbs/show?prefix=Suzuki}{BBS} with Suzuki on Heisenberg doubles. Also her \arXiv{1612.08262}, Kashaev's \arXiv{q-alg/9503005}, Baseilhac's \arXiv{math/0202272} and \arXiv{1101.3440}, and Baseilhac-Benedetti's \arXiv{1101.1851}.

\vskip -1.5mm % -2mm is too much
\needspace{12mm} % 11mm is not enough.
\parpic[r]{\includegraphics[width=1.6in]{figs/LopezNeumann-2a.png}}
\entry{230425b} L\'opez-Neumann@[K-OS], twisted Drinfel'd doubles: Let $H$ be a f.d.\ $\bbN$-graded Hopf algebra and let $\theta(h) = t^{|h|}h$, $h\in H$. Virelizier (2000): for each $n\in\bbZ$ let $D_n=H^*\otimes H$ with multiplication as on the right. This is NOT a Hopf algebra (if $n\neq 0$) but there is a ``coproduct'' $\Delta_{n,m}\colon D_{n+m}\to D_n\otimes D_m$ and antipode $S_n\colon D_n\to D_{-n}$ satisfying graded versions of Hopf axioms.
%\entry{230425b} L\'opez-Neumann@[K-OS], twisted Drinfel'd doubles:
%\par\includegraphics[width=\linewidth]{figs/LopezNeumann-2.png}

\entry{230425a} {
\def\Rep{{\operatorname{Rep}}}
L\'opez-Neumann@[K-OS]: $X\in\Rep(DH)$ iff $X\in\Rep(H)$ together with a collection of {\it half-braidings} $\{\sigma_{Y,X}\colon Y\otimes X \to X\otimes Y\}_{Y\in\Rep(H)}$. Thus, $\Rep(DH)$ is the {\it Drinfel'd Centre} of $(\Rep(H),\otimes)$ (centre as for monoids, but it's a monoidal category!). It is a braided monoidal category.
}
%\entry{230425a} L\'opez-Neumann@[K-OS]:
%\par\includegraphics[width=\linewidth]{figs/LopezNeumann-1.png}

\entry{230711} What's the canopoly envelope of Khovanov's Frobenius algebra as generated by $\{\HSaddleSymbol, \odot_\pm, \otimes_\pm\}$?

\entry{230612b} For matrices $A$, $B$, is $\{\lambda\colon \exists v\ Av \!=\! \lambda Bv \!\neq\! 0\}$ always finite? (No; take $A,B\colon\bbR^2_{x,y}\to\bbR$ with $A=x+y$, $B=x$. Then with $v=(1,\lambda-1)$, $Av=\lambda Bv=\lambda\neq 0$.)

\entry{230609} {\tt AlexanderUsingDehn.nb} in \href{http://drorbn.net/AcademicPensieve/People/Martchenkov}{AP: P--: Martchenkov}. \refentry{240221b}.

\entry{230607} Abstract version of $\det\begin{pmatrix} A & B \\ C & U \end{pmatrix} = \det(A)\det(U-CA^{-1}B)$?

\entry{230606} Alexander: $tA-A^T$. Tristram-Levine: $(1-\omega)A+(1-\bar\omega)A^T = (\bar\omega-1)(\omega A-A^T)$.

\entry{230525} Are there other nearly-linear invariants of tangles, beyond linking numbers and signatures? \refentry{210114a}

\entry{230419} {\bf Sydney, Mar-Apr 2023.} In \href{http://drorbn.net/AcademicPensieve/Annotations/Alekseev_Kawazumi_Kuno_Naef}{AP: A--: Alekseev\_Kawa--:} Annotated AKKN1.
In \href{http://drorbn.net/bbs/show?prefix=Hogan}{BBS: Hogan}: Multiplication, $\exp$, and $\hexagon$ in $\calA^{1p2s}$ (1 pole 2 strands); implementation in \href{http://drorbn.net/AcademicPensieve/People/Hogan/nb/Comp1ss.pdf}{AP: ---: Hogan: Comp1ss.nb}.
In \href{http://drorbn.net/bbs/show?prefix=Dancso}{BBS: Dancso}: ``weakly parenthesized'' tangles in a $\text{\it PDS}$, searching for $\atdiv$ in strand doubling $\calA^{2p1s}\to\calA^{2p2s}$ (more likely KV2 is in pole doubling $\calA^{1p1s}\to\calA^{2p1s}$, in degree $1$), the pentagon in $\calA^{2p2s}$ (\refentry{230404}).

\entry{230417} {\bf Proj.} Extend knot colouring to tangles in Zombian language.

\entry{200611a} {\bf Q.} What conditions on $(\bbA,\bbB,\langle\cdot\rangle)$ are enough to make the Drinfel'd double associative? Examples beyond Hopf algebras? Restrict attention to braids?

\entry{230321b} {\bf Proj.} Develop a language to describe $\calA^{/(k+1)\text{-co}}$ in terms of $\FA$/$\FL$. What are the atomic spaces and operations?

\entry{230321a} L\'opez-Neumann and van der Veen, \arXiv{2211.15010}, \href{https://lrobert.perso.math.cnrs.fr/kos.html}{[K-OS]}, \href{http://drorbn.net/AcademicPensieve/People/VanDerVeen/Double_of_k[x].pdf}{email}: A ``twisted double'' that may replace the need to deform the co-product.

\entry{230317b} {\bf Q.} In \href{http://drorbn.net/ld22}{PDS}, is there an expansion for $v$-tangles?

\entry{230317a} {\bf Proj.} In \href{http://drorbn.net/ld22}{PDS}, study emergent $v$-tangles and/or $v_1$-tangles.

\entry{230309} In \href{http://drorbn.net/ld22}{PDS}, is $\calA^{*/*+1}$ isomorphic to ``$\FL$-dirty $\calA$''? Is there a homomorphic $ss$-degree expansion $\calA^*\to\calA^{*/*+1}$? Does the sequence $0 \to \langle\text{wheels}\rangle\otimes\FA \to \calA^{/2}_H \to \FA \to 0$ split?

\entry{230315} In \href{http://drorbn.net/AcademicPensieve/2023-03/nb/AlexanderLogConcavity.pdf}{2023-03: AlexanderLogConcavity.nb}: Testing $\log$-concavity for alternating knots, inspired by  \arXiv{2303.04733} by Hafner, M\'esz\'eros, Vidinas.

\entry{230313} Hiroe, Negami, \arXiv{2303.05770} ``Long-Moody Construction of Braid Representations and Katz Middle Convolution''.

\entry{220712b} Construct homomorphic expansions for tangles in $\text{\it PDS}_n$ $ss$-degree by $ss$-degree starting from $\FG_n\to\FA_n$.

\entry{230227} Equivariant linking numbers:
$\bullet$ Kearton ($\sim$1978): Blanchfield formulas.
$\bullet$ Kojima and Yamasaki (1979): First definition.
$\bullet$ Kricker's and Garoufalidis' \arXiv{math/0105028}: Definition from the Kontsevich Integral.
$\bullet$ Garoufalidis' and Teichner's \arXiv{math/0206023}: Abstract definition (also, indication that $\rho_1$ sees genus).
$\bullet$ Ohtsuki 2007 ``Invariants \ldots Surgery Presentations'': Some computations, probably no PD formula.
$\bullet$ Lescop's \arXiv{1001.4474}, \arXiv{1008.5026}: Abstract definitions (also appearing: $(\log\Delta)'$).
$\bullet$ Friedl's and Powell's \arXiv{1512.04603}: Blanchfield formulas from Seifert surfaces.

\entry{230228a} Negi, Prabhakar, Kamada \arXiv{2302.13244}, twists in v-knots:
\par\includegraphics[width=\linewidth]{figs/TwistedReidMoves.png}

\entry{230217} {\bf Groningen, Jan-Feb 2023.} In \href{http://drorbn.net/AcademicPensieve/Projects/MetaCalculi/nb/UnitaryGamma.pdf}{Projects: MetaCalculi: UnitaryGamma.nb}: A pairing with a Lagrangian property and an $\omega/\bar\omega=\det$ property for $\Gamma$-calculus (single variable). In \href{http://drorbn.net/AcademicPensieve/Talks/MoscowByWeb-2104/nb/Hodge.pdf}{Talks: MoscowByWeb-2104: Hodge.nb}: Hodge infrastructure for $\calA$-calculus; exponentials are preserved, but unfinished unitarity property. Lashing fails at \href{http://drorbn.net/AcademicPensieve/Projects/APAI/nb/Lashings.pdf}{Projects: APAI: Lashings.nb}.

\entry{221118} {\bf Do.} Unify Goldman-Turaev (GT) with equivariant intersection numbers. Understand the action of the braid group on GT (and its expansions). Related to the unitarity of Gassner?

\entry{230213b} {\bf Do.} Digest $S(V)/\Lambda(V)$-exponentiation as ``unfurling''.

\entry{230213a} Re.\ \href{http://drorbn.net/mo21}{{\greektext web}/mo21}, is there a fast zero-test for linear combinations of Fermionic Gaussians?

\entry{230103} {\bf Q.} Are there ``$R$-matrix'' long knot invariants in which a distinction is made between $R_{i<j}$ and $R_{i>j}$? ``Scheduled tangles''?

\entry{230209} {\bf Chal.} Find topological interpretations for QA Alexander formulas. I.e., what do specific matrix entries mean?

\entry{230203b} {\bf Q.} Might it be that determinant formulas for Alexander should depend on a presentation matrix along with a certificate that it presents a torsion module?

\entry{230203a} {\bf Do.} Analyze Alexander determinants for tangles similarly to the signature analysis.

\entry{230201b} In $\Gamma$, is there an infinitesimal version? Are there $\im(\alpha)$ ``classicality'' conditions?

\entry{230201a} In $\Gamma$, for classical tangles $\bar{A}=\calL_0A$ with analytic $\calL_0$. Is $(\bar{A},\bar{\omega})=\calL(A,\omega)$ with analytic $\calL$?

\entry{230118} {\bf Proj.} Write \verb"ValidatePD" and \verb"PlanarQ" for \verb"KnotTheory`".

\entry{230131} In $\Gamma$, truly understand the ``flatness property'' of $\omega$.

\entry{221213} The signed number of ascending crossings is invariant under R2/R3.

\entry{221212} {\bf Q.} Is Alexander a counting of representations?

\entry{221207} Which Gauss diagram formulas vanish on classical knots / tangles? Tanglify ``the linking matrix of a classical link is symmetric''.

\entry{221128} {\bf Q.} Is there a 3D understanding of the balancing of $\Delta(K)$?

\entry{221114} Burton \href{https://drops.dagstuhl.de/opus/volltexte/2020/12183/}{tabulated} knots to 19 crossings.

\entry{221028} Is $g_{\alpha\beta}(K)$ the ?-invariant of a 3-component tangle $K\cup L_{\alpha\beta}$?

%\vskip 0mm % 0mm is too much.
\needspace{3mm} % 3mm is not enough.
\parpic[r]{\scalebox{0.55}{\input{figs/4_1.pdf_t}}}
\entry{221025b} Boyle@Oaxaca: Develop a finite type theory for SNACKs, Strongly Negative AmphiChiral Knots (long knots invariant under $180^\circ$ rotation with vertical flip, as $4_1$). Probable examples in \arXiv{2206.03598}. {\bf Q.} If SNACKs are equivalent as knots, are they equivalent within SNACKs? Are they the same as knots in $\bbR\bbP^3$? (No and no).

%\vskip 0mm % 0mm is too much.
\needspace{21mm} % 20mm is too little.
\parpic[r]{\includegraphics[width=1.8in]{figs/Boyle-Chen-Knot.png}}
\entry{2210a5a} Learn how to compute invariants of knots presented by surgeries. E.g., Boyle's@Oaxaca knot, \arXiv{2207.12593}.

\entry{221007} With $V$ the Burau representation of $\wB_n$, there is a non-trivial splitting of $\bbQ\wB_n\ltimes\bbH(V\otimes V^\ast)\llbracket\eps\rrbracket_{\text{doc}}\to\bbQ\wB_n$. So what?

\entry{220928} {\bf Q.} Given a classical $K$, $\exists ?$ automorphism $\phi\colon\pi(K)=\pi\righttoleftarrow$ s.t.\ $\bar\phi\colon\pi/[\pi,\pi]\cong\bbZ\righttoleftarrow$ is $(-1)$?

\entry{220920} In \href{http://drorbn.net/AcademicPensieve/Projects/RibbonKnots/nb/Rho4Ribbons.pdf}{Projects: RibbonKnots: Rho4Ribbons.nb}: $(p=1-3T+T^2)^2\mid\Delta(11_{n66})$ yet $p\not\mid\rho_1(11_{n66})$. Is there a Fox-Milnor $K$ and a $p$ s.t.\ $p^2\mid\Delta(K)$ yet $p\not\mid\rho_1(K)$? \refentry{220816}

\entry{220712a} A quadratic form $Q$ on $V$ induces $Q'$ on $V/\langle u\rangle$ via $Q'(x,y)=Q(u,u)Q(x,y)-Q(x,u)Q(u,y)$. Best if $Q(u,u)\neq 0$. Leads to a characterization of signatures?

\entry{220821} {\bf Proj.} Find Duflo in Goldman-Turaev.

\entry{220816} {\bf Q.} When the Alexander polynomial factors, is there a reason? (It factors more than Jones). Is every ribbon knot the sum of a w-knot with its mirror, modulo Alexander skein moves?

\entry{220807} {\bf Proj.} Develop a ``universal'' Goldman-Turaev theory: With $G$ a group and $F$ a free group, with $|G|\coloneqq G/\Ad G$ and $|GG|\coloneqq G\times G/\Ad G$ (note the maps $|GG|\to|G|\times|G|$ and $|GG|\to|G|$) there are $b\colon\bbQ|F|\wedge\bbQ|F|\to\bbQ|FF|$ and $\delta\colon\bbQ|F|\to\bbQ|FF|/\text{alt}$. (1) Really? (2) What Lie-bialgebra-like properties hold? (3) Expansions? (4) KV?

\entry{150224b} Fadell-Neuwirth: For $0\!<\!r\!<\!n$, $m\!\geq\! 0$, and $M\!=\!\Sigma^2_{g\geq 1}\!\mid\! D^2$, $1 \!\to\! \PB_{n-r}(M\setminus\underline{m\!+\!r}) \!\to\! \PB_n(M\setminus\underline{m}) \!\to\! \PB_r(M\setminus\underline{m}) \!\to\! 1$ is exact. {\bf Q.}~An infinitesimal version?

\entry{220615b} {\bf Q.} Which maps $\varphi\colon \bbQ F_n\to \bbQ F_n$ preserve the filtration(s)? Just homomorphisms combined with translations? Which decrease the filtration? Riffled splicings with homomorphisms with intersection countings?

\entry{220615a} Does the recovery formula of \href{http://www.math.toronto.edu/~drorbn/Talks/HUJI-1912}{HUJI-1912},
\[ P^{(1)} =
    T \omega  \dot{\omega} - p_1 (T-1)^2/T
    + 2 T \omega  \dot{\omega} a
    + 2 T \omega  \dot{\omega} xy/(1-T),
\]
hold for virtuals?

\entry{220530} Is the $\diamond$-algebra generating function poly-computable?

\entry{220426} Lamm lists ribbons with up to 12 crossings.

\entry{220421} {\bf Proj.} Use AI to rediscover Fox-Milnor.

\needspace{26mm} % 25mm is not enough.
\parpic[r]{\includegraphics[width=0.57\linewidth]{figs/TribracketAxiom.png}}
\entry{220413} Niebrzydowski, Pilitowska, Zamojska-Dzienio \arXiv{1805.07817} ``Knot-theoretic Ternary Groups''. Also, Chavez, Nelson \arXiv{2204.05851} ``$\Delta$-Tribrackets and Link Homotopy''.

\entry{140312} Minsky: Fary-Milnor: $\gamma\colon S^1\to\bbR^3$, $\kappa$ its total curvature. Then $\kappa<4\pi$ $\Rightarrow$ $\gamma$ unknotted.
{\em Pf.} Find a projection direction $p\in S^2$ in which $\gamma$ has $\leq 3$ criticals, hence $2$ criticals, hence $\gamma$ is 1-bridge. As $\dot\gamma$ travels length $\kappa$ on $S^2$, $(\dot\gamma)^\perp$ spans area $<4\kappa$, hence some $p$ is covered $<4$ times. $\Box$\ \ {\em Further proofs.} Petrunin, Stadler \arXiv{2203.15137}. {\em Generalization.} $2\pi(\text{bridge number})=(\text{infimal }\kappa)$.

\entry{220321} {\bf Q.} What are the ADO (Akutsu-Deguchi-Ohtsuki) invariants?

\entry{220302} Giroux, Goodman, \arXiv{math/0509555}: fibered links all come from plumbing.

\entry{201225b} {\bf Proj.} Develop poly-dimensional braid representations via ``Burau/Gassner homology'' (cf.\ \href{http://drorbn.net/AcademicPensieve/2020-07}{2020-07}: \href{http://drorbn.net/AcademicPensieve/2020-07/nb/HeisenbergPerturbations.pdf}{HeisenbergPerturbations.nb}).

\entry{220128} Miller: \href{https://math.berkeley.edu/~kmill/notes/alexander_poly_2020_11_14.pdf}{All the Ways I Know to Define the Alexander Polynomial}.

\entry{220102} {\bf Dream.} (1) An extension theorem for Heisenberg modules. (2) All quantum invariants arise in this way.

\entry{211225} {\bf Q.} A quantum mechanical interpretation of coloured Jones?

\entry{211222b} Owens' \arXiv{2112.10706} has a list of sliceness criteria.

\entry{211222a} Owens and Swenton's ``An Algorithm to Find Ribbon Disks for Alternating Knots'', \arXiv{2102.11778} and \href{https://cat.middlebury.edu/~mathanimations/klo/ribbondisks/}{here}, has programs, data, and stats. \refentry{210517}

\entry{211221} {\bf Q.} Mirror and glue a slice disk $(\bbR^2_+,S=\bbR)\subset(\bbR^4_+,\bbR^3)$ to get a w-knot $K=\bbR^2\subset\bbR^4=\bbR^4_+\cup\bbR^4_-$. Is $A(S)=|A(K)|^2$?

\entry{211206} Signatures: I know neither the domain (tangles as what algebraic structure?) nor the range (as algebro-geometric object, without evaluation in $\bbR$/$\bbC$). Are signatures compatible with strand doubling? What's the multi-variable version?

\entry{211114} Is there a universal way to express the quotient MVA$\to$Alexander?

\entry{211014} Aizawa, Harada, Kawaguchi, Otsuki, \arXiv{math/0612781}: All Link Invariants for Two Dimensional Solutions of YBE.

\entry{210930} In Ruppik's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}: Any epimorphism $\pi_1(\Sigma_g)\to FG_g$ is realized by a handlebody.

\entry{210923} Roland: $\bbe^{\hbar x}$ makes sense in $(\bbZ/p)[\hbar]/(\hbar^p)$.

\entry{210918} Compact $\Leftrightarrow$ Every open cover that's closed under pair unions contains the full set.

\entry{210916} {\bf Do.} Implement and double the Taft Hopf algebra (e.g.\ sec.~4 of Montgomery, Schneider, ``Skew derivations of finite-dimensional algebras and actions of the double of the Taft Hopf algebra''). Compute invariants. Compare with \arXiv{2103.01081} by Feng, Hu, and Li, \arXiv{1805.10340} by Cline, and \href{https://math.rice.edu/~notlaw/ZacharyCline-dissertation.pdf}{Cline's thesis}.

\entry{210908} Farley's \arXiv{2109.02815}: The Planar Pure Braid Group, $\Gamma_n\coloneqq\pi_1(\bbR^n\setminus\{\text{triple intersections}\})$.

\entry{210905b} The thickening $\{\text{signed bipartite planar graphs}\}\to\calK$ is surjective. Is there a local theory? Finite-type? Virtual?

\entry{210905a} Checkerboard / Alexander knots: Reidemeister theory? Algebraic structure? Finite-type theory? Relations with almost classical? Signatures? Other invariants?

\entry{141113b} Boden: A v-knot is ``Almost Classical (AC)'' if it is homologically trivial on a surface. Equivalent to ``image in $\calK(\bigcirc_v\upcap\!\!_w)$ splits''? Is there a FT theory for AC knots? Is there ``$H^1$ of the carrying surface'', an invariant of v-knots? \refentry{231105a}

\entry{210806} Y-C.\ Tang: $\operatorname{u-width}(K)=6>4=\operatorname{v-width}(K)$ for                       $K\in\{9_{18},9_{27},10_{24},10_{25},10_{42},10_{44}\}$.

\entry{210802} Write Duflo in {\bf DoPeGDO} language.

\entry{170316} \href{https://en.wikipedia.org/wiki/Wigner-Weyl_transform}{Wikipedia: Wigner-Weyl transform}, \arXiv{hep-th/0011137} (\& within): Gaussians compose hyperbolically,
$\exp \left (-{a } (q^2+p^2)\right )\star
  \exp \left (-{b} (q^2+p^2)\right ) = {1\over 1+\hbar^2 ab}
  \exp \left (-{a+b\over 1+\hbar^2 ab} (q^2+p^2)\right )$.

\entry{210414} Signatures: The ``standard'' \`a la Tristram--Levine, {\tt KnotTheory`}, Goeritz (e.g. Ozsv\'ath-Stipsicz-Szab\'o), Kashaev's \arXiv{1801.04632}, A.~Conway's \arXiv{1903.04477}. Also Cimasoni-Conway \arXiv{1507.07818} and Merz' \arXiv{2104.02993} on additivity defects of signatures of braid closures.

\entry{210106} The ``Iterates Completion'' $\calM$ of a {\bf Vect}-enriched category $\calC$ with an ideal $\calI$: adjoin ``iterates'' $a^\star\coloneqq(1-a)^{-1}$ for endomorphic $a\in\calI$.
Commutes with additive completion, with positive coefficients:
$\begin{pmatrix}a&b\\c&d\end{pmatrix}^\star = \begin{pmatrix}
  (a+bd^\star c)^\star & \!a^\star b(d+ca^\star b)^\star \\
  d^\star c(a+bd^\star c)^\star\! & (d+ca^\star b)^\star
\end{pmatrix}$!
If $\calC$ is monoidal, contains $\det(1-A)^{-1}$ for $A\in\calI$?
Relations beyond $a^\star b(d+ca^\star b)^\star = (a+bd^\star c)^\star bd^\star$ with $a,(b|c),d\in\calI$ at
$\ \xymatrix{\bullet \ar@`{p+(-8,8),p+(-8,-8)}^a \ar@/^/[r]^(0.35){b} & \bullet \ar@/^/[l]^(0.35){c} \ar@`{p+(8,8),p+(8,-8)}_d}\ $?
Also/just $(a+b)^\star=b^\star(ab^\star)^\star=a^\star(ba^\star)^\star$?
Canonical forms for morphisms? In poly-time?
Carries Alexander?
\href{http://drorbn.net/AcademicPensieve/2021-01/nb/Det3x3.pdf}{2021-01: Det3x3.nb}, \refentry{181222b}, ``weighted automata'', ``rational series''.

\entry{151216} {\bf Proj.} Braidors \& weak associators: $B=\Phi^{012}R_u^{12}\Phi^{-021}$,
\[ B^{012}B^{02,1,3}B^{023} = B^{01,2,3}B^{013}B^{03,1,2}. \]
Extensibility / uniqueness in $\calA^u$, $\sder$, $\Gamma$? A KV/WKO3 variant? Are they enough to quantize Lie bialgebras? To solve YB?

\entry{210525} {\bf TIL} (Kuperberg on FB, \href{http://drorbn.net/AcademicPensieve/2021-05/nb/FubiniCounterexample.pdf}{2021-05: FubiniCounterexample.nb}) $\int_{[0,\infty]^2}\frac{(x-y)dxdy}{(1+(x-y)^2)^2}$, a nice Fubini counterexample.

\entry{210517} Manolescu@BIRS: Sherry Gong has a program to find ribbons for (ribbon) knots. \refentry{211222a}

\entry{210512} {\bf Q.} What is docility for maps between polynomial rings?

\entry{210506} $\pi_1(\SL_2(\bbR))=\bbZ$.

\entry{210429} On the Blanchfield pairing: \mo{7411}.

\entry{210408} Lobb's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}, ``Four-Sided Pegs Fitting Round Holes Fit All Smooth Holes'', inspires plotting $\text{M\"ob}=\operatorname{Conf}_2(S^1\subset\bbC)\to\bbC^2$ via $(z,w)\mapsto((z+w)/2,(z-w)^2)$ to get a circle-boundary M\"obius band in $\bbR^3$.

\entry{210329} {\bf Q.} What values take singular Gaussian Fermionic integrals?

\entry{210318c} {\bf Chal.} Interpret Leclerc's ``On Identities Satisfied by Minors of a Matrix'' in terms of Fermionic Gaussians.

\entry{210318b} Grinberg's book, ``Notes on the Combinatorial Fundamentals of Algebra''.

\entry{210316} {\bf Q.} Why are Fermionic Gaussian compositions so similar to Bosonic ones?

\needspace{8mm} % 7mm is not enough.
\parpic[r]{$\displaystyle
\begin{CD}
  \begin{array}{c|ccc}
    \omega & c & S \\
    \hline
    c & \alpha & \theta \\
    S & \psi & \Xi
  \end{array}
  \hspace{-2.2mm}@>\Gamma::\tr_c>\mu\coloneqq 1-\alpha>\hspace{-1.8mm}
  \begin{array}{c|cc}
    \mu\omega & S \\
    \hline
    S & \!\Xi+\psi\theta/\mu\!
  \end{array}
\end{CD}$}
\entry{180821b} I don't understand the MVA (and yet it's there). E.g., does the Halacheva meta-trace have a tensorial representation \refentry{180909a}? Why is it independent of the component left open?

\entry{210211a} $\Lambda^\ast(X^\ast\cup Y)$ with $\deg X^\ast=-1$, $\deg Y=1$ is a contraction algebra with
  $c_{x,y}\coloneqq\bbe^{\partial_\eta\partial_x}\colon\Lambda^\ast(X^\ast\cup Y)\to\Lambda^\ast((X-x)^\ast\cup(Y-y))$
hence
  $\calA(X)\coloneqq\Lambda^\ast(X^\ast\cup X)_0$
is a traced meta-monoid with $m^{xy}_z\coloneqq c_{xy}\act(\zeta\to\xi,y\to z)$ and $\tr_x(A)\coloneqq c_{xx}$.
Halacheva ($\sim$): Contains $\Gamma$ (w/ fixed colours) via
  $\Upsilon_X\colon(\omega,M)\mapsto\omega\bbe^{\sum\xi M_{\xi y}y}$.
Predict $\calA$ from $\Gamma$? Interpret $\calA$ in $ybax$? Related to super-algebras? Raise $\calA$ to meta-Hopf? Understand $\im(\Upsilon)$?

\entry{210217} {\bf Do.} In {\bf DoPeGDO} for $ybax$, do $a$ first?

\entry{201217a} {\bf Proj.} A concise proof of Alexander / Markov (including an $n^{3/2}$ complexity bound and an implementation). {\bf Q.} Is there a framed Markov theorem? Lambropoulou, Rourke ``Markov's Theorem in 3-Manifolds'': A ``Markov theorem'' using only $L$-moves:
\includegraphics[width=\linewidth]{figs/L-Moves.png}
Also: Sundheim, Traczyk, Morton, Birman, Birman-Brendle.

\entry{210211c} {\bf Do.} Super-{\bf DoPeGDO}.

\entry{210211b} {\bf Do.} $\calA^w$ and super-Lie-algebras.

\entry{190314b} Wherefore $\left(\sum_{n\geq 0}\tr\Lambda^n A\right)\exp\left(\sum_{n>0}\frac{(-1)^n}{n}\tr A^n\right) = 1$? \href{http://drorbn.net/AcademicPensieve/2021-02/nb/rdet.pdf}{2021-02/rdet.nb}: use for quick $\det$ computations in rings in which non-zero integers are invertible. \refentry{210130}

\entry{200703a} {\bf Q.} An Archibald calculus that includes strand doubling? A common generalization of Archibald- and $\Gamma$-calculus? \refentry{200804}, \refentry{210211a}

\entry{210204} Conway's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}, ``Knotted Surfaces with Infinite Cyclic Knot Group'': a topological classification of surfaces $\Sigma$ with $\pi_1(\Sigma^c)=\bbZ$ in a simply-connected 4-manifold.

\entry{210130} Itai: Computing $\det$ over a ring $R$ in poly-time: compute $\det(1-\epsilon(1-A))$ in $R\llbracket\epsilon\rrbracket$ by Gaussian elimination and formal inversions. But it's a polynomial in $\epsilon$! Now set $\epsilon=1$. \refentry{190314b}

\entry{210114a} Feller's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}: the fractional Dehn twist coefficient, the unique homogeneous quasimorphism $\omega\colon B_n\to\frac1n\bbZ$ with defect $\sup_{g,h\in G} |f(gh)-f(g)-f(h)| = 1$ s.t.\ $\omega(\Delta^2)=1$, $\omega(B_{n-1})=0$. Invariant under conjugation, probably under strand doubling. Also Malyutin, ``Twist Number of (Closed) Braids''.

\entry{210128b} {\bf Do.} Use $\OU$ to compute $\calA^w(\bigcirc_n)$. \refentry{200906}

\entry{210128a} Hom's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}, ``Infinite Order Rationally Slice Knots'': Rationally slice $\coloneqq$ bounds a smooth disk in a $\bbQ HS^4$. {\bf Thm.} Using Heegaard-Floer, there is a $\bbZ^\infty\oplus(\bbZ/2)^\infty$ in rationally slice knots modulo concordance. Levine 69': algebraic concordance group $\coloneqq$ $\{$Seifert forms$\}/$(metabolic, vanishes on half-dimensional subspace) $\equiv \bbZ^\infty\oplus(\bbZ/2)^\infty\oplus(\bbZ/4)^\infty$.

\entry{210124b} Manturov's philosophy: ``If something is wrong, it's because it's not drawable on the plane. This must be because of a homological obstruction, which leads to a parity, leading to projections, brackets, coverings.''

\entry{210124a} Manturov (zoom): his ``Parity and Projection from Virtual Knots to Classical Knots'' has a combinatorial proof of Kuperberg's theorem.

\entry{210121} Khovanov's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}, ``Bilinear Pairings and Topological Theories'': ``near TQFTs'' regressed from arbitrary invariants; especially in 2D.

\entry{210114b} {\bf TIL.} ``Digit ratio''.

\entry{210107} The denominators miracle: \newline
\includegraphics[width=\linewidth]{figs/DenominatorsMiracle.png}
\newline\null\hfill$=a+((1-c)de+(1-b)fg+dmg+fne)/((1-b)(1-c)-mn)$
\newline\null\hfill$=a+(db^\star e+fc^\star g+db^\star mc^\star g+fc^\star nb^\star e)(mb^\star nc^\star)$

\entry{200917} {\bf Do.} Zipping in the 1PI context. Convert diagram from ``external assembly'' to ``merging of completed''.

\vskip -1mm % -2mm is too much.
\needspace{18mm} % 17mm is too little.
\parpic[r]{
  \def\h{{$\frac12$}} \def\a{{$\frac{\alpha}{2}$}} \def\g{{$\frac{\gamma}{2}$}}
  \scalebox{0.82}{\input{figs/ZippingTwice.pdf_t}}
}
\picskip{4}
\entry{201225a} \href{http://drorbn.net/AcademicPensieve/Projects/BabyDoPeGDO/nb/ZippingTwice.pdf}{Zipping twice}:
$\alpha = \frac{t^2}{1-\left(a+\frac{b^2}{1-c}\right)} = \frac{t^2(1-c)}{(1-a)(1-b)-b^2}$,
$\beta = t\frac{sb}{1-c}\frac{1}{1-\left(a+\frac{b^2}{1-c}\right)} = \frac{tsb}{(1-a)(1-b)-b^2}$,
$\gamma = \frac{s^2b^2}{(1-c)^2}\frac{1}{1-\left(a+\frac{b^2}{1-c}\right)} + \frac{s^2}{1-c} = \frac{s^2b^2+s^2((1-a)(1-c)-b^2)}{(1-c)((1-a)(1-c)-b^2)} = \frac{s^2(1-a)(1-c)}{(1-c)((1-a)(1-c)-b^2)} = \frac{s^2(1-a)}{(1-a)(1-c)-b^2}$.
\vskip 1mm

\vskip -1mm % -2mm is too much.
\needspace{8mm} % 7mm is too little.
\parpic[r]{$\begin{pmatrix}1&a&b\\a&e&\!\!1-et\!\!\\b&\!\!1-e\!\!&et\end{pmatrix}$}
\entry{201230} \href{http://drorbn.net/AcademicPensieve/People/VanDerVeen/nb/SingularGamma.pdf}{People: VanDerVeen: SingularGamma.nb}: $S_{ab}$ (on right) is a general singular point in $\Gamma$-calculus, with $\left.S_{ab}\right|_{e=1}=R_{ab}$ and $\left.S_{ab}\right|_{e=t^{-1}}=R^{-1}_{ba}$.

\vskip -1mm % -2mm is too much.
\needspace{1mm} % 0mm is too little.
\parpic[r]{\scalebox{1.15}{\input{figs/MarkovRiddle.pdf_t}}}
\entry{201217b} {\bf Q.} If $\beta_{1,2}\in B_n$ then $\beta_1\sigma_n\beta_2\sigma_n^{-1}$ and $\beta_1\sigma_n^{-1}\beta_2\sigma_n$ have the same closure. How are they related by Markov moves?

%\needspace{8mm} % 0mm is too little.
\parpic[r]{\scalebox{1}{\input{figs/MarkovSolution.pdf_t}}}
Sol'n from Birman-Brendle \arXiv{math/0409205}, Fig.~13: ``Markov'' the red strands on the right.

\entry{201214b} Polyak: $\{\text{planar diagrams}\}/(R1r,R2b,R3b)$ describes knots via braids, but by counting counterclockwise cycles in the oriented smoothing, are more than knots.

\entry{201214a} {\bf Q.} What are $\{\text{planar curves}\}/(R1l,R1r,R2b,R3b)$?

\entry{201209} Bolan: $\sqrt{2}^{\log_2 9}\in\bbQ$.

\entry{201117} {\bf Q.} Are there GPV formulas for tangles and links?

\entry{201109} There is a ``crossing change'' construction of Seifert surfaces, and a construction starting from an immersed bounding disk.

\entry{201112} Piccirollo's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}, ``A Users Guide to Straightforward Exotica'': has a calculus for handle decompositions of 4-manifolds.

\entry{201105} Dynnikov's \href{http://homepages.warwick.ac.uk/~masgar/Seminar/current_seminar.html}{talk}, ``An Algorithm for Comparing Legendrian Knots'': Legendrian knot: in $\ker(xdy+dz)$. Have Reidemeister theory. Related to grid diagrams. Topological meaning?

\entry{201023} {\bf TIL.} Merge ``only'' in ``she told him that she loved him''.

\entry{181222b} For
$A \coloneqq \begin{pmatrix}a&b\\c&d\end{pmatrix} = \begin{pmatrix}1&b\bar{d}\\0&1\end{pmatrix} \begin{pmatrix}a-b\bar{d}c&0\\0&d\end{pmatrix} \begin{pmatrix}1&0\\\bar{d}c&1\end{pmatrix} \in M_{(p+q)\times (p+q)}$,
$|A|=|d|{|a-b\bar{d}c|}$,
and
$A^{-1} = \begin{pmatrix} (a-b\bar{d}c)^{-1} & -(a-b\bar{d}c)^{-1}b\bar{d} \\ -\bar{d}c(a-b\bar{d}c)^{-1} & \bar{d}+\bar{d}c(a-b\bar{d}c)^{-1}b\bar{d} \end{pmatrix}$,
with $\bar{d} \coloneqq d^{-1}$.
\href{http://drorbn.net/AcademicPensieve/2019-03}{2019-03}, \refentry{190312}, \refentry{210106}, Bareiss' ``\href{https://www.ams.org/journals/mcom/1968-22-103/S0025-5718-1968-0226829-0/S0025-5718-1968-0226829-0.pdf}{Sylvester's Identity\ldots}''.

\entry{181222a} If $A\in M_{n\times n}$ and $\omega=|A|$, then each entry of $A^{-1}$ has denominator $\omega$, so expect $|A^{-1}|\,\propto\,\omega^{-n}$. Yet $|A^{-1}|=\omega^{-1}$.

\entry{201011} The Steinberg relations between elementary matrices: $e_{ij}(\lambda)e_{ij}(\mu)=e_{ij}(\lambda+\mu)$, $[e_{ij}(\lambda),e_{jk}(\mu)]=e_{ik}(\lambda\mu)$ for $i\neq k$, and $[e_{ij}(\lambda),e_{kl}(\mu)]=1$ for $i\neq l$ and $j\neq k$.

\entry{201006} {\bf Q.} Let $I_{v\downarrow w}\coloneqq\ker(\PvB\to\PwB)$ and let $\calA_{v\downarrow w}\coloneqq\prod I_{v\downarrow w}^m/I_{v\downarrow w}^{m+1}$. What is $\calA_{v\downarrow w}$? Are there ``expansions''? Are there ``Burau/Gassner expansions''?

\entry{200925} {\bf Riddle} (Matthew Bolan) $\exists?$ cont.\ $f\colon\bbR\to\bbR$ s.t.\ $f\circ f=\cos$?

\entry{200216} Direct proof that $\uB_n\hookrightarrow\vB_n$ (cf.\ Gaudreau, \arXiv{2008.09631}): For $w=\prod_{\alpha=1}^l\sigma_{i_\alpha j_\alpha}^{s_\alpha}$ define ``depths'' $(d_{k,\alpha})_{1\leq k\leq n,0\leq\alpha\leq l}$ inductively by $d_{k0}=k$ and for $\alpha>0$, $d_{k\alpha} = d_{k,\alpha-1}+s_\alpha(\delta_{ki_\alpha}-\delta_{kj_\alpha})$. Drop from $w$ every $\sigma_{i_\alpha j_\alpha}^{s_\alpha}$ for which $s_\alpha\neq d_{j_\alpha,\alpha-1}-d_{i_\alpha,\alpha-1}$ (well defined on $\vB_n$!). Iterate. The result is a retraction $\vB_n\to\uB_n$.

\entry{200906} With $\frakg=sl_{2+}^0=\langle y,b,a,x\rangle/([a,x]=x,\,[a,y]=-y,\,[x,y]=b,\,[b,-]=0)$, $\calU(\frakg)_\frakg$ is freely generated by $\yellowm{\{b^ka^n\}_{k\leq n}}$, also $\yellowm{\{y^ka^nx^k\}_{k,n\geq 0}}$, for $[a,y^la^mx^n]=(n-l)y^la^mx^n$ forces $l=n$ and $xa=(a-1)x$ implies $xf(a)=f(a-1)x$ so $[x,f(a)] = {(f(a-1)-f(a))x}$ so with $a^{(k)}\coloneqq\binom{a+k}{k} = {(a+1)(a+2)\cdots(a+k)/k!}$ and $a^{(-1)}\coloneqq 0$, $\left[x,a^{(k)}\right]=-a^{(k-1)}x$ so $\left[x,y^na^{(k)}x^{n-1}\right] = nby^{n-1}a^{(k)}x^{n-1}-y^na^{(k-1)}x^n$. Thus in $\calU(\frakg)_\frakg$, $\yellowm{by^nx^n=0}$ and $\yellowm{y^na^{(k)}x^n = n!b^na^{(k+n)}}$. So $\{b^ka^n\}$ generates and $b^{n+1}a^n=0$. The same relations also follow from $[y^{n-1}a^{(k)}x^n,y] = -y^na^{(k-1)}x^n + nby^{n-1}a^{(k)}x^{n-1}$, and these are all the relations in $\calU(\frakg)_\frakg$. \refentry{230915}

\entry{180909a} The meta-trace (simpler before quantization?): At $k=0$, $\tr_m=\bbE\left(\beta_m b_m,\, \frac{\calA_m(1-B_m)}{\calA_m-1}\xi_m\eta_m\right)$. Then ruled by $C_{abc} \xrightarrow{d^2} C_{12} \xrightarrow{d^0} C_0$ via $d^1 = \mu^{12}_0-\mu^{21}_0$ and $d^2 = (\sigma^a_1\mu^{bc}_2 + \text{cyc. perm.})$, where $\mu^{ij}_k = (\alpha_k\to\alpha_i+\alpha_j,\, \xi_k\to\calA_j\xi_i+\xi_j,\, \eta_k\to\eta_i+\calA_i\eta_j,\, \beta_k\to\beta_i+\beta_j-\xi_i\eta_j)$. More generally,
$\tr_m=\bbE\left(r\alpha_m\beta_m/\hbar+s\beta_m b_m+t\alpha_ma_m,\, \frac{\calA_m^{1-r}(\calA_m^r-B_m^s)}{\calA_m-1}\xi_m\eta_m\right)$. \href{http://drorbn.net/ap/2018-08/nb/Trace.pdf}{Pensieve: 2018-08: Trace.nb}, \refentry{180821b}, \refentry{230915}.

\entry{200827} Baby {\bf DoPeGDO} in \href{http://drorbn.net/AcademicPensieve/2020-07/nb/HeisenbergPerturbations.pdf}{2020-07: HeisenbergPerturbations.nb} and \href{http://drorbn.net/AcademicPensieve/People/VanDerVeen/nb/TimidHeisenbergRGeneralForm@.pdf}{People: VanDerVeen: TimidHeisenbergRGeneralForm@.nb}.

\entry{190304} Exponential zipping: With $[f]_\lambda\coloneqq\bbe^{\lambda\partial_z\partial_\zeta}f = \left\langle f|_{z\to z+z',\,\zeta\to\zeta+\zeta'}\right\rangle_{\lambda;z'\leftrightarrow\zeta'}$ (so $\left.\langle f\rangle_\lambda=[f]_\lambda\right|_{z=\zeta=0}$) and $P_\lambda\coloneqq\log[\bbe^f]_\lambda$, have $P_0=f$ and $\partial_\lambda P_\lambda = (\partial_z\partial_\zeta P_\lambda)+(\partial_z P_\lambda)(\partial_\zeta P_\lambda)$. {\bf Proj.} Extend {\bf DoPeGDO} to {\bf EDDO} (Exponentiated Docile Differential Operators). \href{http://drorbn.net/AcademicPensieve/2019-03/}{2019-03}: requires the ``wake equation'' $\partial_\lambda W^j = W^i\partial_{z^i}W^j$.

\entry{200811} {\bf Q.} In $CU$, if $Q$ is quadratic and $P$ is docile, are $\log\bbO(\bbe^Q)$ quadratic and $\log\bbO(\bbe^P)$ docile?

\entry{160312} {\bf Proj.} Truly understand $Kh(K)=1\Rightarrow K=1$. If $Kh(D)=1$, constructively reduce $D$ to 1. Kronheimer, Mrowka \arXiv{1005.4346}, ``Khovanov homology is an unknot-detector''.

\entry{200807} {\bf Q.} Is there a unique factorization $K=\kappa\,\#\,K'$, with $K,K'$ virtual knots, and $\kappa$ maximal classical?

\entry{200806} {\bf Q.} Is there a link invariant {\it UMVA} such that $\text{\it UMVA}(L)$ dominates all the {\it MVA}s of satellites of $L$? \refentry{200703a}

\entry{200804} Costantino-Le \arXiv{1907.11400}: $\calO_{q^2}(\text{\it SL}(2))$ in skein theory.

\entry{20721} {\bf Q.} Is there a unique factorization $T=\beta'T'$, with $T,T'\in\calT_n$, and $\beta'\in\calB_{2n}$ ``maximal''? What is $\calT_n/\calB_{2n}$?

\entry{200703b} Kassel-Turaev: $\Sigma$ a punctured disk with basepoint $d$, $\varphi\colon\pi_1(\Sigma)\to G$ a surjective homomorphism, $(\tilde{\Sigma},\tilde{d})$ the cover of $(\Sigma,d)$ corresponding to $\ker\varphi$. Then $\tilde{H}\coloneqq H_1(\tilde{\Sigma},\tilde{d})$ is a left module over $\bbZ G$ of free rank $\operatorname{rk} H_1(\Sigma)$.

\entry{200625} Beliakova@\href{https://lrobert.perso.math.cnrs.fr/kos.html}{KOS}: ``Partial trace property''.

\entry{200618} Meusburger@\href{https://lrobert.perso.math.cnrs.fr/kos.html}{KOS}: Mapping class groups presentations by Gervais, Penner, Bene. ``Pivotal Hopf monoids''. How much of her work is OU / sutured 3-manifolds? {\bf Q.} Are there virtual versions of other mapping class groups? Will they have simpler presentations, like $\PvB$ is simpler than $\PB$?

\entry{200611b} D.~Long: Given $\rho\colon AB_n=F_n\rtimes B_n\to\End(V)$ constructs $\rho^+$ acting on $(\aug\bbZ F_n)\otimes_{\bbZ F_n}V = \bbZ^n\otimes_\bbZ V$.

\entry{200523} {\bf TIL.} $E_1(x)\coloneqq\int_x^\infty\frac{\bbe^{-t}dt}{t}\sim\sum_{n\geq 0}\frac{(-)^nn!}{x^n}$. Also $\int_0^\infty\frac{\bbe^{-x}dx}{1+\eps x} \sim \sum_{n\geq 0}n!(-\eps)^n$.

\needspace{18mm} % 17mm is not enough.
\parpic[r]{\includegraphics[width=25mm]{figs/PiccirillosKnot.png}}
\entry{200521b} Piccirillo's knot \arXiv{1808.02923} requires $Kh(55\text{xings})$.

\entry{200521a} {\bf TIL.} Naisse: ``graded monoidal category''.

\entry{200520} Is the gr of acyclic tangles acyclic arrow diagrams? What's the right algebraic structure?

\vskip -2mm % -3mm is too much.
\needspace{16mm} % 15mm is too little.
\parpic[r]{\hspace{-3mm}\def\arraystretch{0}\begin{tabular}{r}
  \includegraphics[width=1.061in]{figs/IceBreakers.jpg} \\
  \scriptsize\href{http://www.waterfrontbia.com/ice-breakers-2019-presented-by-ports/}{www.waterfrontbia.com}
\end{tabular}\hspace{-2mm}}
\entry{180406} {\bf Proj.} A volume $V$ yarn-ball knot has $\sim V^{4/3}$ xings. Can compute linking numbers in $\sim\!V$ time. $\pi_1$ has $\sim\!V$ generators/relations. Is the degree of Alexander is bounded by $\sim\!V$? Same for $\rho_1$? How big is a KTG presentation? How high the genus? The hyperbolic volume? The degree of Jones?

\entry{170401} {\bf Proj.} over-then-under ``$\OU$-Tangles''. Closed under compositions; (v-)braids are $\OU$; non-braid $\OU$-tangles? Relations in $\OU$? In $\calA^\OU$? Not all tangles are $\OU$? Alexander properties; v-version. Associators in $\calA^u\cap\calA^\OU$: Constructible? Sufficient for EK? Relations with Chterental's ``virtual curve diagrams''? {\bf Q.} Is there a quotient-completion-extension $\calT\to\tilde{\calT}$ of $\calT=\{\text{tangles}\}$ s.t.\ in $\tilde{\calT}$ every tangle is $\OU$ and s.t.\ all Reshetikhin-Turaev invariants factor through $\tilde{\calT}$? The w reduction (with pacifiers?)? Does $\OU/R1,R2\hookrightarrow\vT$?

\entry{200326} Meadow: commutative ring with unit with $x\mapsto x^{-1}$ s.t.\ $(x^{-1})^{-1}=x$ and $x(xx^{-1})=x$.

\entry{200320} Audoux, Bellingeri, Meilhan, Wagner in \arXiv{1507.00202}: Is $\PvB\to\PvT$ injective?

\entry{200205} The Hopf axiom $S\ast I=I\ast S=\eta\eps$ is too strong, so for involutive-Hopf-algebra invariants R2b $\Rightarrow$ R2c. \refentry{180909b}.

\entry{200204a} $\hat{\frakg}\coloneqq\frakg[t^{\pm 1}]\oplus\langle c\rangle$ with $[c,-]=0$, $[t^na,t^mb]=t^{n+m}[a,b]+\delta_{n+m}n\langle a,b\rangle c$.

\entry{170126d} {\bf Wanted.} A finite-dimensional representation of $gl_{n,+}^\eps$.

\entry{191127} {\bf Q.} A ``representation'' theory $\frakg_+^\eps\to gl_{n,+}^\eps$? Weyl actions?

\entry{191119} Kopparty on Berlekamp-Welch: The values of a degree $n$ polynomial $p$ are given on a set with $10n$ points with $n$ lies. Can recover $p$ in poly time! Used in ``Reed-Solomon Codes''.

\entry{191112b} {\bf Do.} GDO for all Lie algebras.

\entry{191112a} {\bf TIL.} PIT (Polynomial Identity Testing) is BPP (use random evaluations on scalars and Zariski density) but unknown if in P.

\entry{191107b} {\bf Def.} A $Q$-module: An assignment $Q\mapsto\calP_Q$ taking $\calS(V^{\otimes 2})\to\text{Set}$, along with maps $\calP_{Q_1}\times\calP_{Q_2}\to\calP_{Q_1+Q_2}$ and $\calP_Q\to\calP_{[F\colon Q]}$ for $F\in\calS((V^\ast)^{\otimes 2})$. Interesting examples?

\entry{191107a} {\bf Do.} Put co-Poisson-Hopf-algebras in {\bf DoPeGDO}.

\entry{191021} \bbs{Dror}{191021}{022146}:
\newline\includegraphics[width=\linewidth]{figs/Dror-191021-022146.jpg}
\newline$\bullet$ Requires $[r,a_i\otimes 1 \!+\! 1\otimes a_i] \!=\! 0$.
$\bullet$ Geometric meaning for $\calA^{v+}$?
$\bullet$ A diagrammatic quantization in $\calA^{v+}$?
$\bullet$ An $\OU$ version?

\entry{191104} Whirling (\href{http://drorbn.net/ap/2019-11}{2019-11}): $W\colon\begin{pmatrix}\Xi&\phi\\\theta&\alpha\end{pmatrix} \mapsto \frac{1}{\alpha} \begin{pmatrix}1&-\theta\\\phi&\alpha\Xi-\phi\theta\end{pmatrix}$ satisfies $W^n(A)=A^{-1}$ for $A\in M_{n\times n}$. Why won't denominators explode?

\entry{191030} A direct sum Lie algebra can degenerate to a non direct sum.

\entry{191029} {\bf Do.} 6T and TC solutions in 3D are too weak for Alexander.

\entry{141114a} {\bf Proj.} Short paper on ``crossing the crossings'', $\Zhe\colon\vK_n\to\wK_{n+1}$: definition, invariance, u-neutrality, associated graded. Domination of Manturov, Bardakov, Boden, Brandenbursky. Relation with Medina-Revoy. \refentry{141107}.

\entry{191020} {\bf Do.} {\bf DoPeGDO} directly with $sl_2^\epsilon$ (or why it can't be done).

\entry{190808} {\bf Do.} For $D\in\calA^u$, recover $\calT_{\frakg+}(D)$ from $\calT_\frakg(D)$.
Recover the $sl_{2+}$ invariant from the $sl_2$ one.

\entry{191001} ``The $n$-Category Cafe is the left adjoint of the inclusion functor of obscure mathematics into all mathematics'' (credits on file).
% Credits: Outlook/191001

\entry{190914} Are there ``homological virtual knots'', where only the homology of a carrying surface matters?

\entry{190906} Pulmann-\v{S}evera @\arXiv{1906.10616}:
In $\calC$ a $BMC$, $[HA(\calC)]\equiv[\text{lax monoidal nerves }N\colon BrCom\to\calC]$.
In $\calC$ an $R$-$iBMC$, $[PHA(\calC)]\equiv[\text{lax monoidal nerves }N\colon iCom(R)\to\calC]$.
Drinfel'd: ($\calC$ an $R$-$iBMC$) $\leadsto$ ($\calC_\hbar^\Phi$ a $BMC$).

\entry{190722} {\bf Do.} Develop the $\OU$/$\bbA\bbB R$ narrative to a solution of $R2c$. Find a linear-complexity embedding of tangle theory into diagrams mod braid-like moves.

\entry{190723} Friedl, Powell @\arXiv{1907.09031}: if a knot $J$ is homotopy ribbon concordant to $K$ then $A(J)\mid A(K)$. An AKT view?

\entry{190711} {\bf Q.} Wherefore the Heisenberg Double $H(A)$? Kashaev (95'): The Drinfel'd double $D(A)\subset H(A)\otimes H(A)^{op}$.

\vskip -1.5mm\needspace{9mm} % 8mm is too little, 12mm is too much.
\parpic[r]{$\begin{pmatrix}
  g|_U & 0 & 0 \\
  0 & g|_V & v \\
  0 & 0 & 1
\end{pmatrix}$}
\entry{190709} Naef: If $gl(U)\supset G\lefttorightarrow V$, a faithful representation of $(g,v)\in G\ltimes V$ in $U\oplus V\oplus\bb1$:

\entry{190612} Boot up to \v{S}evera: $\bullet$ Implement $RI=\bbe^{t_{12}/2}$. $\ldots$

\entry{190606} {\bf Needed.} A precedent for ``two-stage Gaussian integration''.

\vskip -1mm
\needspace{21mm} % 20mm is not enough.
\parpic[r]{$\xymatrix@C=8pt@R=8pt{
  \calT \ar@/^/[drr]^{Z^v} \ar[dr]|-{\tilde{\calT}} \ar@/_/[ddr]_{Z_{URT}} &
    \raisebox{10pt}{\text{\footnotesize filtered}} &
    \raisebox{10pt}{\text{\footnotesize graded}} & \\
  & \tilde{\calA} \ar[r] \ar[d] & \calA^v \ar[d] &
    \hspace{-15pt}\text{\footnotesize universal} \\
  & \calU_\hbar(\frakg) \ar[r]^<>(0.5){Z_\frakg} & \calU(\frakg)\llbracket\hbar\rrbracket &
    \hspace{-12pt}\text{\footnotesize specific} \\
  &  &  &
}$}
\picskip{6}
\entry{171001} What the world should look like.
$Z_\frakg$: A representation theoretic construction? A homological construction? A soft construction from $Z^v$? A torsor? $GT/GRT$?
$\tilde{\calA}$: a universal $\calU_\hbar(\frakg)$. An a priori meaning?
$\tilde{\calT}$: A universal extension/quotient of $\calT$. Rotational virtual tangles? A quotient thereof? $v$-Claspers? A closure of $\OU$? \refentry{190105a}.

\entry{190603} Kofman in \href{https://www.math.csi.cuny.edu/~ikofman/growth_voldet_slides.pdf}{Da Nang}: The Vol-Det Conjecture: For a hyperbolic alternating link $K$, $\operatorname{Vol}(K)\leq 2\pi\log\det(K)$.

\entry{190530} Porti in \href{http://mat.uab.cat/~porti/Alex.pdf}{Da Nang}: For a hyperbolic $K\subset S^3$ and $|\zeta|=1$, $\lim_{N\to\infty}N^{-2}\log|\Delta_k^{\rho_N}(\zeta)|=\operatorname{vol}(S^3\setminus K)/4\pi$, with $\rho_N$ the $N$-dim representation of $\text{\it SL}_2(\bbC)$.

\entry{190523} \bbs{Itai}{190523}{131710}+: $L^p$ inequalities: $\|u\|_q\leq C_{p,d}\|\nabla u\|_p$ in $\bbR^d$ with $\frac1q=\frac1p-\frac1d$; $\|u\|_p\leq\|u\|_{p_0}^{1-\theta}\|u\|_{p_1}^\theta$.

\entry{190113} The non-linear Schr\"odinger eqn: $\bbi\partial_t\psi = -\frac12\Delta\psi - |\psi|^{p-1}\psi$. \bbs{Itai}{190523}{125238}: conserves $\int|\psi|^2$ and $\int\left(\frac12|\nabla\psi|^2-\frac{2}{p+1}|\psi|^{p+1}\right)$.

\vskip 0mm\needspace{6mm} % 5mm id too small, 11mm is too large.
\parpic[r]{\scalebox{1}{\input{figs/DDFormula.pdf_t}}}
\entry{180619} Does the Drinfel'd double generate all ``suppressed-cycle diagrams''?

\entry{190509} Snyder, Tingley @\arXiv{0810.0084}: $T$ with $R\!=\!T_1^{-1}T_2^{-1}\Delta_{12}T$. In {\bf DoPeGDO}?

\entry{190501b} Merkulov @\arXiv{1904.13097}: ``Grothendieck-Teichm\"uller Group, Operads and Graph Complexes: a Survey''.

\entry{190501a} Livingston @\arXiv{1504.03368}: ``Doubly Slice Knots with Low Crossing Number''. An AKT description? \refentry{181218b}

\entry{190425b} {\bf DoPeGDO}$_2$: Quadratics are of weight precisely 2, interactions of weight at most $2$, with $\wt(x,y,\xi,\eta,a,b,\alpha,\beta,\epsilon)=(1,1,1,1,2,0,0,2,-2)$.

\entry{190425a} Darn\'e @\arXiv{1904.10677}: w-braids up to homotopy.

\entry{190417} {\bf Riddle.} If a box of sides $(b_i)$ is contained in a box of sides $(a_i)$, then $\sum b_i\leq\sum a_i$. Khesin's and Itai's Sol'ns in \%.
%\par{\bf Sol'n} (Khesin). An $\epsilon$-neighborhood of the $B$ box is contained in an $\epsilon$-neighborhood of the $A$ box, so by comparing volumes (in 3D, but it generalizes):
%\begin{multline*}
%  \frac{4\pi\epsilon^3}{3}+3\pi\epsilon^2\sum b_i+2\epsilon\sum_{i<j} b_ib_j+\prod b_i \\  \leq
%  \frac{4\pi\epsilon^3}{3}+3\pi\epsilon^2\sum a_i+2\epsilon\sum_{i<j} a_ia_j+\prod a_i
%\end{multline*}
%Now compare powers of $\epsilon$ when $\epsilon$ is large.
%\par{\bf Sol'n} (Itai). If $v_i$ are the sides of the inner box and the coordinate system is parallel to the sides of the outer box, then $\sum\|v_i\|_2\leq\sum\|v_i\|_1$, and the latter is the sum of dimensions of the outer box by a coord-by-coord inspection.

\entry{190414} With Ens.
$\bullet$~Establish a cluster$\Leftrightarrow$DK dictionary.
$\bullet$~Do syzygy operators always eliminate the image of some ``error operator'', like $\tilde{d}^1\act\tilde{d}^2=0$ in \href{http://www.math.toronto.edu/~drorbn/papers/GT1/GT1.pdf}{papers/GT1}?

\entry{190412b} {\bf Chal.} Mix braidors and solvable approximation.

\entry{190412a} {\bf Q.} Is there an interesting ``braidor-liberator'' algebra?

\entry{190407} DCA $\coloneqq$ Directed Circuit Algebra $=$ Symmetric Strict Spherical Category with singly-generated set of objects.

\entry{190409} Khovanskii @\arXiv{1904.03341}: ``One Dimensional Topological Galois Theory''.

\entry{190404} {\bf Do.} Implement Habiro-Massuyeau \arXiv{1702.00830}, section~9.1:\newline
\includegraphics[width=\linewidth]{figs/HM-P55.png}

\entry{190324} In rec-tangles: rectangle = handle = hand = sleeve = protected zone = input of op. Is there a 3D description?

\entry{190325} {\bf TIL.} Mathematica's \href{https://reference.wolfram.com/language/Notation/guide/NotationPackage.html}{Notation} package.

\entry{190322} Fiedler @\arXiv{1902.06091}: ``A refinement of the first Vassiliev invariant can distinguish the orientation of knots''.

\entry{171205} (Approx.) On $H^{\ast\text{\it cop}}\otimes H$ with $R=Id=\rho\otimes r$ (summed), $\int\phi\otimes x\coloneqq \langle\phi\bar{\rho}\mid xr\rangle$ is an integral.
{\bf $\frac12$Pf.} $x_1\int\phi\otimes x_2
= x_1\langle\phi\bar{\rho}\mid x_2r\rangle
= x_1r^ar^b\langle\phi\bar{\rho}\bar{\rho}^a\rho^b\mid x_2r\rangle
\sim x_1r_1r^b\langle\phi\bar{\rho}\rho^b\mid x_2r_2\rangle
\sim (xr)_1r^b \langle\phi\bar{\rho}\mid (xr)_3\rangle \langle\rho^b\mid (xr)_2\rangle
\sim (xr)_1(\overline{xr})_2 \langle\phi\bar{\rho}\mid (xr)_3\rangle
= \langle\phi\bar{\rho}\mid xr\rangle = \int\phi\otimes x$. {\red Verify!} Attempt in \href{http://drorbn.net/AcademicPensieve/Projects/SL2Portfolio2/nb/DoubleIntegration.pdf}{Projects/SL2Portfolio2/DoubleIntegration.nb}.

\entry{190314a} {\bf Do.} A graphical calculus for {\bf DoPeGDO} and {\bf DoPeGDO}$_2$.

\entry{190321} After Rushworth: Is Jones on $\bbQ$HS a polynomial? A skein relation? A Kauffman bracket? Categorifies? \refentry{140116}

\entry{190312} {\bf Def.} $\omega\|A$ means $\forall k\,\omega^{k-1}|\Lambda^k(A)$. {\bf Ex.} With $\omega=|A|$, $\omega\|(\omega B+\adj(A))$ (tested \href{http://drorbn.net/AcademicPensieve/2018-12}{2018-12}). {\bf Q.} An effective $\omega\|A$ certificate strong enough to certify the example? \refentry{181222b}

\entry{190310b} Tree generation: $\partial_\lambda T_\lambda = (\partial_z T_\lambda)(\partial_\zeta T_\lambda)$. Legendre transform: $(Lf)(\eta) \coloneqq \crit_y(\eta y-F(y)) = \eta y_0-F(y_0)$, with $dF_{y_0}=\eta$.

\entry{190310a} KV in $\Gamma$ in \href{http://drorbn.net/AcademicPensieve/Projects/MetaCalculi/}{Projects/MetaCalculi}:\newline
\includegraphics[width=\linewidth]{../MetaCalculi/KVinGamma2.png}

\entry{190225} ``Set-theoretically-induced delusions of greatness''.

\entry{131203} \v{S}evera quantization, \arXiv{1401.6164}: Given a BMC $\calD$ {\magenta (with Manin $(\partial, \frakg, \frakg^\star)$, set $\calD\coloneqq\calU(\partial)$-$\Mod^\Phi$)}, a co-braided co-algebra $(M,\Delta\colon M\to M^2,\epsilon\colon M\to 1_\calD)$ in it {\magenta ($M\coloneqq\calU(\frakg)=\calU(\partial)/\calU(\partial)\frakg^*$)}, a second BMC $\calC$ {\magenta (Vect)}, a functor $F\colon\calD\to\calC$  {\magenta ($F(X)\coloneqq X/\frakg X$)} and a comonoidal structure $c$ (natural  $c_{X,Y}\colon F(XY)\to F(X)F(Y)$ and $c_1\colon F(1_\calD)\to 1_\calC$ respecting braiding and associativity) so that
\[ \begin{CD}F(XMY)@>F(1\Delta 1)>>F(XMMY)@>c_{XM,MY}>>F(XM)F(MY)\end{CD} \]
\[ \text{and}\quad\begin{CD}F(M)@>F(\epsilon)>>F(1_\calD)@>c_1>>1_\calC\end{CD} \]
\vskip 2mm
are isomorphisms {\magenta(the clear $c_{X,Y}\colon XY/\frakg(XY)\to(X/\frakg X)(Y/\frakg Y)$)}, construct a Hopf algebra structure on $H\coloneqq F(M^2)$:
\[ \Delta_H\colon\
  \begin{CD}
    F(M^2)@>F(\Delta\Delta)>>F(M^4)@>F(1R1)>>F(M^4)@>c_{M,M}>>F(M^2)^2,
  \end{CD}
\]
\[ m_H\colon\qquad
  \begin{CD}
    F(M^2)^2 @<c_{M^2,M^2}\circ F(1\Delta 1)<\raisebox{2mm}{\Large $\sim$}< F(M^3) @>F(1\epsilon 1)>> F(M^2),
  \end{CD}
\]
\[ S_H\colon\qquad\qquad
  \begin{CD}
    F(M^2) @>F(R)>> F(M^2).
  \end{CD}
\]

Set also $G\colon X\mapsto F(MX)$ {\magenta($G\colon X\mapsto\frac{\calU(\frakg)X}{\frakg(\calU(\frakg)X)}$)}, the ``twist''.

-- Is $H$ the symmetry algebra of something? -- In the non-quasi case, can we reconstruct $\calU(\frakg)$ from the category of $\partial$-modules? -- In the abstract context, what is the relation between $H$ and $M$? -- How does this restrict to AT/AET in the commutative case?

-- $H$ pairs with the quantization of $\frakg^\star$? \v{S}evera in \href{http://drorbn.net/dbnvp/LD15_Severa-2.php}{LD15/II}: No.

\vskip -2mm
\needspace{5mm} % 4mm is not enough.
\parpic[r]{\scalebox{0.95}{\input{figs/TwoTubeSurgery.pdf_t}}}
\entry{190221} {\bf Do.} Understand ``two tube surgery'':

\entry{190205} {\bf Q.} When/why does the universal Verma module see all invariants?

\entry{190124} {\bf TIL.} ``Groupoid algebras'' are ``weak Hopf algebras''.

\entry{181106a} Guo@Hefei: Algebras: $\bullet$ Rota-Baxter: Associative with unary $P$ with $P(f)\ast P(g) = P(f\ast P(g))+P(P(f)\ast g))$. Think $P(f)=\int f$, $f\ast g=\int f'g'$. $\bullet$ Dendriform; pre-Lie; averaging; diassociative.

\entry{190105b} {\bf Q.} Why is there a Cartan involution for classical $sl_{2,\epsilon}$ (multiplicative co-multiplicative $\theta$ preserving $r+r^{21}$)? Is there a quantum analog?

\entry{190105a} {\bf Q.} If $A_h$ is the quantization of a Lie bialgebra $\fraka$, is there always a multiplicative expansion $A_h\to\calU(\fraka)\llbracket h\rrbracket$? \refentry{171001}.

\entry{190104} {\bf Wanted.} Examples around ``$\left.\partial_x(x^{-1})\right|_{x=0}$ is undefined, yet $\left.\bbe^{\xi\partial_x}(x^{-1})\right|_{x=0}=\xi^{-1}$''.

\entry{190102} Etingof-Schiffmann 2.2.2: $\langle a,x\rangle/([a,x]=x)$ is also a Lie bialgebra with $\delta(a,x)=(a\wedge x,0)$.

\vskip 0mm\needspace{11mm} % 10mm is too little.
\parpic[r]{\scalebox{0.95}{\input{figs/ZippingTheorem.pdf_t}}}
\entry{180629} The Zipping Thm (verification \href{http://drorbn.net/AcademicPensieve/2018-12/nb/ZippingTheorem.pdf}{2018-12}). If $P$ has a finite
$\zeta$-degree and $\tilde{q}$ is the inverse matrix of $1-q$:
$(\delta^i_j-q^i_j)\tilde{q}^j_k=\delta^i_k$, then
\[ \left\langle
    P(z_i,\zeta^j)\bbe^{c+\eta^iz_i+y_j\zeta^j+q^i_jz_i\zeta^j}
  \right\rangle
  \!=\! |\tilde{q}|\left\langle
    \left.P(z_i,\zeta^j)\bbe^{c+\eta^iz_i}\right|_{z_i\to\tilde{q}_i^k(z_k+y_k)}
  \right\rangle
\]
\[
  = |\tilde{q}|\bbe^{c+\eta^i\tilde{q}_i^ky_k}
    \left\langle P\left( \tilde{q}_i^k(z_k+y_k),\zeta^j+\eta^i\tilde{q}_i^j \right) \right\rangle.
\]
%\begin{multline*} \left\langle
%    P(z_i,\zeta^j)\bbe^{c+\eta^iz_i+y_j\zeta^j+q^i_jz_i\zeta^j}
%  \right\rangle_{(\zeta^j)} \\
%  = \det(\tilde{q})\left\langle
%    \left.P(z_i,\zeta^j)\bbe^{c+\eta^iz_i}\right|_{z_i\to\tilde{q}_i^k(z_k+y_k)}
%  \right\rangle_{(\zeta^j)}.
%\end{multline*}
{\bf Do.}
$\bullet$ Sort $\langle\cdot\rangle_{\text{int}} \leftrightarrow \langle\cdot\rangle \leftrightarrow \mathrlap{\int}{\scriptstyle G}$.
$\bullet$ Sort denominators.

\entry{181218b} Freedman: an AKT description of slice knots? \refentry{190501a}

\entry{181218a} {\bf Q} (following Boden via Gaudreau). Is the crossing number of a virtual link equal to that of its irreducible representative?

\entry{181202} {\bf TIL.} Vibration modes of mugs (Tadashi Tokieda).

\entry{181201} {\bf Do.} A better narrative for $\doubletree$.

\entry{181123} {\bf Proj.} Clasp number $k$ knots: AKT-definable? Alexander properties? \refentry{161009}.

\vskip -1.5mm \needspace{5mm} % 4mm is not enough.
\parpic[r]{\scalebox{1.22}{\input{figs/PeggedXing.pdf_t}}}
\entry{181119} {\bf Do.} EK in terms of pegged tangles:

\entry{181116} Chang, Cui @\arXiv{1710.09524} relate Kuperberg/Turaev-Viro-Barrett-Westbury with Hennings-Kauffman-Radford/Witten-Reshetikhin-Turaev.

\entry{181104} AC$\Rightarrow$Zorn: Assume by contradiction that in $(X,<)$ every chain $C$ has a (chosen) *strict* bound $M(C)$, and let $\calW\coloneqq\{W\subset X\colon W\text{ well ordered},\,\forall x\in W\,M(\{w\in W\colon w<x\})=x\}$. Then $\bigcup\calW$ is a maximal element of $\calW$ (effort here), contrary to the existence of $M(\bigcup\calW)$. (The key: transfinite constructions have a ``maximal extent'' $\calM$; here leading to a contradiction. AC is not needed for $\calM$, yet it has a busy beaver feel.)

\vskip -2mm\needspace{6mm} % 5mm is not enough!
\parpic[r]{$\xymatrix{
  & E \ar[d]_p \\
  W \ar[r]^{g_t} \ar[ur]^{\forall h_0\,\exists h_t} &
  B
}$}
\entry{150609d} Fibrations $p\colon E\to B$ on right. Any $X\xrightarrow{\phi}Y$ is $X\xrightarrow{i_0}E_\phi\xrightarrow{p_1}Y$ with $i_0$ a homotopy equivalence and $p_1$ a fibration. Here $E_\phi\sim\{(x\in X,\gamma:[0,1]\to Y)\colon \gamma(0)=\phi(x)\}$; $i_0(x)=(x,\bar{x})$, $p_1(x,\gamma)=\gamma(1)$, and the ``homotopy fiber'' is $p_1^{-1}(y)=\{(x,\gamma)\colon\gamma(0)=\phi(x),\,\gamma(1)=y\}$. E.g., for $\Emb(\bbR\hookrightarrow\bbR^3)\hookrightarrow\Imm(\bbR\to\bbR^3)$, the homotopy fiber is framed knots.
\par \vskip -2mm
\needspace{12mm} % 11mm is too little.
\parpic[r]{$\xymatrix@C=12mm{
  A \ar[d]^f \ar[dr]^{g_t} & \\
  X \ar[r]^(0.4){\forall h_0\,\exists h_t} &
  Z
}$}
Cofibrations $f\colon A\to X$ on right (e.g.\ cones, $A\to CA$). Any $X\xrightarrow{\phi}Y$ is $X\xrightarrow{i_0}M_\phi\xrightarrow{p_1}Y$ with $i_0$ a cofibration and $p_1$ a homotopy equivalence.

\entry{181106b} \href{https://cgp.ibs.re.kr/~gabriel/}{Drummond-Cole} knows dimensions of the BiAlg PROP.

\vskip -1.5mm\needspace{0mm}
\parpic[r]{\scalebox{1.22}{\input{figs/HopfRules.pdf_t}}}
\entry{181028} Wherefore these relations, with $n\in 2\bbZ$?

\entry{181017a} ``Topological Expansionism'': ``Quantum Topology'' is a mix of topology, algebra, representation theory, and quantum field theory. I will explain how to expand the territory of topology within that mix at the expense of representation theory and algebra.

\vskip -1.5mm\needspace{12mm} % 11mm is too short.
\parpic[r]{\scalebox{1}{\input{figs/Swirl.pdf_t}}}
\entry{181022} {\bf Do.} Understand the swirl:

\entry{181019} Bruguieres, Virelizier @\arXiv{math/0505119}: ``Hopf diagrams and quantum invariants''.

\entry{181017b} Hass, Thompson, Tsvietkova @\arXiv{1809.10996} ``alternating links have at most polynomially many Seifert surfaces of fixed genus'': In AKT language?

\entry{181014} {\bf Do.} Show that KV is a full triangularity equation, as Duflo is triangularity in co-invariants.

\entry{181013} Khanin: $\sum 1/n^2=\pi^2/6$ by comparing coefficients of $x^2$ in $\frac{\sin x}{x} = \prod(1-x^2/\pi^2n^2)$, itself true by comparing roots and constant terms.

\entry{181009} de Mesmay, Rieck, Sedgwick, Tancer @\arXiv{1810.03502}: realizations of some logic circuits in KO $\Rightarrow$ many KT problems are NP-hard.

\vskip -1.5mm\needspace{17mm} % 16mm is not enough.
\parpic[r]{\scalebox{0.8}{\input{figs/KeyDD.pdf_t}}}
\entry{160611} Majid's Primer \S 8.1: the quantum double $\calD A \coloneqq {A^{\ast,\text{\it op}}\otimes A}$~with $(\phi a)(\psi b) \!\coloneqq\! \langle Sa_1,\psi_1\rangle \langle a_3,\psi_3\rangle (\psi_2\phi)(a_2b)$ (``{\it op}'' for multiplication). What problem does it solve? Two layers of wrong: 1. $R$ isn't in the result so it shouldn't be in the motivation. 2. The result is degree-non-decreasing, the formula should be the same. \refentry{180725}. {\bf Q.} It's an image construction. A kernel one?

\entry{181001} {\bf Riddle} (Dylan). Warden to 100 prisoners: I've chosen a permutation $\pi$ of your names and tomorrow I will place each of you in an isolated room with 100 boxes storing $\pi$. You will each get to open 50 boxes and each must open their ``own'' box. Maximize the probability of success.  Sol'n \href{http://drorbn.net/AcademicPensieve/2018-10/}{2018-10}.

\vskip -2mm
\needspace{20mm} % 19mm is not enough.
\parpic[r]{\scalebox{1}{\input{figs/DrinfeldElement.pdf_t}}}
\entry{180424} \bbs{VanDerVeen}{170622}{113622}, verification \href{http://drorbn.net/AcademicPensieve/2017-06/nb/Doubling.pdf}{Doubling.pdf}, in $R_{12}^{-1}=S_2(R_{12})$ conventions. The Drinfel'd's cuap: $u\coloneqq R_{12}\act S_1\act m^{21}$, $v^2\coloneqq S(u)u$, $C\coloneqq uv^{-1}$. Properties: $S^2(z)=u^{-1}zu$ (pf?), $uS(u)=S(u)u$ is central. Issues: Invariance property of $C$? $R$ is in the total-rotation-$0$ subspace, and all operations preserve it. How can they generate $C$? Perhaps it's the distinction between the pre- and post-doubling $S$? \refentry{180724}.

\entry{180910} ``Locally Euclidean Knotted Objects'', leKO. {\bf Thm.} leKO $\Leftrightarrow$ $\bbR$-rotation vKO; and $\bbZ$-rotation vKO $\Leftrightarrow$ leKO with all measurable rotation numbers in $\bbZ$.

\entry{180905} {\bf Riddle} (Itai). Fairly select 1 in 1,001 in 2 tosses of some fair $p$-dice, $p<1,000$ prime. Sol'n in \%.
% Hint: 311=7*11+11*13+7*13 is prime.
% Also, following Dylan/180929:
% Sort@Select[Tuples[Range[1, 12], 3].{7 11, 11 13, 13 7}, (# < 1000) && PrimeQ[#] &]
% {311,479,619,647,661,727,751,773,787,829,839,853,857,881,883,919,941,947,971,983,997}

\entry{180909b} Green, Nichols, Taft: ``Left Hopf Algebras'', $S\ast I=\eta\epsilon$, $I\ast S\neq\eta\epsilon$. Also Lauve, Taft @\arXiv{0908.3718}: ``[Left $\SL_q(n)$]''. \refentry{200205}.

\entry{180904b} {\bf Riddle} (Tsimerman). How many not-necessarily-fair coins to fairly select $1$ of $n$ in finitely many tosses? Sol'n in \%.
%Answer: 2. One 1/2 coin to be tossed approximately 2log_2(n) times, and one
%1/n coin to be tossed at most once.
%
%Details: Find p and positive a and b such that 2^p=an+b(n-1); this is always
%possible if 2^p>n(n-1), but sometimes possible for smaller p. Toss the 1/2
%coin p times to get a fair number smaller than an+b(n-1). If it less than
%an, take it mod n and you got a fair number smaller than n. If not, toss the
%1/n coin once. If it comes out on the "1" side, output n. Otherwise use the
%b(n-1) you have left from before to output something less than n-1.
%
%Better answer: 1. If p+q=1 are the head/tail probabilities, partition the
%2^d monomials of the form p^kq^(d-k) in (p+q)^d into (n-1) manifestly equal
%subsets and one smaller subset A which nevertheless contains the unique p^d
%monomial. At p=1 A is always chosen, at p=1/2 it is chosen less than the
%others. By IVT at some p the odds are even.

\entry{180830} {\bf Riddle} (\href{https://blog.tanyakhovanova.com/2018/08/another-cool-coin-weighing-problem/}{Khovanova}) Among 14 coins, 7 weigh $a$ each, and 7 weigh $b<a$ each. You're told which is which. Confirm this with three uses of a balance scale. (``leverage'')

\entry{180904c} What's ``tangle planarity'' in evaluation diagrams?

\entry{180904a} Dylan's sutured 3-manifolds: $\partial M=R_+\cup_\sigma R_-$. Balanced: $\chi(R_+)=\chi(R_-)$ (per component?). Taut: if $\Sigma\subset M$ with $\partial\Sigma=\sigma$ then $\chi(\Sigma)\leq\chi(R_\pm)$ (wherefore?). Equivalence: add or remove a $D_1\times D_2$ with suture $\{0\}\times D_2$ along $S^0\times D_2$ or along $D_1\times S^1$. Marking: by multiple disjoint arrows in $R_+$ and in $R_-$, with ends on $\sigma$. Gluing with $90^\circ$ rotation! Contains $m$, $\Delta$, $S$ if $S^2=1$. {\bf Q.}~Faithfulness and completeness? Separation algorithm? $S^2\neq 1$? {\bf Proj.}~Write ``3-manifolds and the Drinfel'd double construction''.

\entry{180821a} The open Hopf $\OpenHopf$ maps dual co-invariants to invariants.

\entry{180827} {\bf Riddle} (Ido). On an $n$-vertex directed graph with letter-marked vertices, every length $2^n$ word can be formed with walks. Prove that the same is true for all words. Sol'n \href{http://drorbn.net/AcademicPensieve/2018-08/}{2018-08}.

\entry{180818} Kirk's unitarity: $X$ is the complement of a pure tangle in $D^2\times[0,1]$, $X_i\coloneqq X \cap (D^2 \times \{i\})$ for $i=0,1$. Choose a generic $U(1)$ representation $\alpha\colon H_1(X;\bbZ)=\bbZ^n \to U(1)$ with $\alpha(m_k)=t_k$.
%$\alpha$ extends to an embedding $\bbQ(t_k) \to \bbC^\ast$.
Then $\alpha|_{X_0}=\alpha|_{X_1}$. Since $t_k \neq 1$, the cohomologies (with coefficients in $\alpha$) of meridians and tubes in $\partial X$ vanish.

By Mayer-Vietoris the restriction map
\[ H^1(\partial X;\alpha)\to H^1(X_0;\alpha)\oplus H^1(X_1;\alpha)\quad (*) \]
is an isomorphism. By the long exact sequence $H^1(X_i, \partial X_i;\alpha)\cong H^1(X_i;\alpha)$. Thus the non-degenerate skew-Hermitian (ndsH) cup product $H^1(\partial X;\alpha)^2 \to H^2(\partial X;\bbC)$ decomposes using (*) as $\operatorname{diag}(A,-A)$ with $A$ the ndsH inner product $\yellowm{H^1(X_0, \partial X_0;\alpha)}^2 \to \bbC$, as the $X_i$'s are disjoint with opposite orientations in $\partial X$.

{\bf Claim. } $x^T A y = g(x)^T A g(y)$ for $g$ the Gassner representation.

{\bf Pf.} 1. The image $H^1(X)\to H^1(\partial X)$ is a Lagrangian $L$ (Poincar\'e duality).
2. The two composites $H^1(X)\to H^1(\partial X)\to H^1(X_i)$ on the summands in (*) are isomorphisms (Le Dimets, also [KLW]).

Using 1 and 2, $x\in H^1(X_0)$ has a unique lift to $(x, g(x))\in L$. If $x,y\in H^1(X_0)$, as $L$ is Lagrangian,
\[ 0= (x ,g(x)) \cup (y, g(y)) = x \cup y - g(x) \cup g(y). \]
(the cross terms vanish because of the $0$s off diagonal.) but $x \cup y = x^T A y$ and $g(x) \cup g(y) = g(x)^T(-A) g(y)$, hence $x^T A y= g(x)^T A g(y)$ \hfill$\Box$

\entry{180815} Blair, Sack @\arXiv{1801.00230}: the tangle category is Karoubi complete: $f^2=f$ $\Rightarrow$ $f=gh$ with $hg=1$. {\bf Q.} v,w?

\entry{180809} Meta-monoids aren't equivalent to monoid objects in a monoidal category: $m^{ij}_k[S]\colon M_{S\sqcup\{i,j\}}\to M_{S\sqcup\{k\}}$ isn't induced from $m^{ij}_k[\emptyset]$. Yet if $M$ is a monoid object in a symmetric strict monoidal $(\calC,\otimes,\bb1)$, then $M_S\coloneqq\mor(\bb1,M^{\otimes S})$ is a meta-monoid.

\entry{180812} {\bf Riddle} (Ido). In an $\underline{n}$-cards game of war, can the sides jointly ensure finiteness from any initial position? {\bf Q.} Effective? Poly-time?

\vskip -1mm\needspace{8mm} % 7mm is not enough.
\parpic[r]{\scalebox{0.56}{\input{figs/2RingsPuzzle.pdf_t}}}
\entry{170928} {\bf Riddle} (Chterental, May 2014). Get left to right moving only blue.

\entry{180811} Tangloids, Medusas. {\bf Q.} Expansions? Intermediate to u and v, so implied by neither; yet implies $\calU(\frakg)$.

\entry{150610} Andrews-Curtis Conjecture: balanced group presentations differ by Nielsen trans.: $(g_i)\to(g_{\sigma i})$, $g_i\to g_i^{-1}$, $g_i\to g_ig_j$. Myasnikov$\times 2$, Shpilrain @\arXiv{math/0302080}: potential counterexamples.

\entry{180806} Schwinger-Dyson is translation invariance of the path integral measure, written using post-integration handles:
\[ 0 \!=\! \int\calD\phi\, \partial_\phi\bbe^{\phi\cdot Q\phi/2+V(\phi)+J\cdot\phi}
  \!=\! \left(Q(\delta_J)+(\partial_\phi V)(\delta_J)+J\right)Z(J).
\]

\vskip 0mm\needspace{5mm} % 4mm is not enough
\parpic[r]{\scalebox{1}{\input{figs/mHa.pdf_t}}}
\entry{180803} {\bf Q.} What axioms befall meta-Hopf-actions?

\entry{180722} $\mu\colon\{\text{DK-Associators}\}\to\text{SolKV}$ is injective as $\mu\act(-\to\{\sder-\text{Associators}\})$ is the identity modulo wheels (also Furusho, Schneps, \href{http://drorbn.net/dbnvp/LD16_Enriquez.php}{Enriquez@LD16}). Surjectivity? \refentry{180721a}.

\entry{180725} {\bf Wanted.} A limited-foresight narrative for (want a stitching- and cabling-compatible invariant of u-tangles) $\Longrightarrow$ (look at $am$, $bm$, $R$, $P$, rotational virtual knots, and the meta-Drinfel'd double procedure). \refentry{160611}.

\entry{180724} (w/ Dylan). $(l+)$-kink$=(r+)$-kink in {\it RVT} implies $s=\text{\footnotesize$\overset{\circlearrowleft}{\bot}$}$ in $\calA^{\text{\it rvt}}$. \refentry{170520}, \refentry{180424}.

\entry{180721a}  {\bf Dream} (w/ Zsuzsi). $d^3\gamma\pi d^2f=0$ when $d^n=\sum_{k=0}^{n+1}(-)^kd^n_k$ the co-Hochschield differential, $\pi\colon\sder_n\to\FL_{n-1}$ the projection on $x_n$-degree 1, $\gamma\colon\FL_{n-1}\to\sder_n$ the Lie morphism with $x_i\mapsto t_{in}$, and with $f\in\sder_2$. Leads to an algebraic construction of DK associators? \refentry{180722}.

\entry{180721b} $\FL(V\oplus W)\cong\FL(\FA(V)\otimes W)\oplus\FL(V)$ like $\FA(V\oplus W)\cong\FA(\FA(V)\otimes W)\oplus\FA(V)$.

\entry{180721c} {\bf Conj} (w/ Zsuzsi). $\sder_n\cong\oplus_{k=1}^n\FL_k^{\text{palindromic}}$.

\entry{180716} {\bf Proj.} Direct proofs of $\uT\hookrightarrow\vT$ and $\uT\hookrightarrow\wT$.

% $\calZ_\lambda\colon \xymatrix{B \ar@/^/[r] \ar[r] \ar@/_/[r] & A}$
\entry{180708} With $\calZ_\lambda= \{(Z\colon B\to A=\gr B)\colon\gr Z=\lambda^{\deg}\}$, have $Z_0\in\calZ_0$ by $b\mapsto[b]_{I}\oplus 0\ldots$. Hence $\GRT\to\calZ_0$. When injective? Surjective? Bijective for $B=\PaB$! For $\epsilon^2=0$, $\calZ_\epsilon\neq\emptyset$ iff there is $\beta_\kappa\in B_\kappa$ for every kind $\kappa$, such that for every op $\rho$, $\rho(\beta\ldots\beta)-\beta\in I^2$ (set $Z_1(b)=[b]_I\oplus\epsilon[b-\beta]_{I^2}\oplus 0\ldots$). In that case, $\GRT\to\calZ_\epsilon$. Injective? Surjective? Always $\calZ_\epsilon\to\calZ_0$; a bijection for $\PaB$.

\entry{180615} {\bf Q.} For the BMC crowd, are tangles a Hopf algebra variant?

\entry{180611} {\bf Q.} Does $t=\epsilon a-\gamma b$ have a topological meaning?

\entry{180528b} Not every v-knot has a Seifert surface. Are there ``Seifert v-tangles''? Is this related to unitarity and to Fox-Milnor?

\entry{180515} {\bf Q.} Wherefore the zip algebra, $\langle z_n\rangle_{n\geq 0}$ with $z_mz_n = \sum_{k=0}^{\min(m,n)}k!\binom{m}{k}\binom{n}{k}z_{m+n-k}$? Representation: $z_n\mapsto\hat{x}^n\partial_x^n$. Isomorphic to $\bbZ[y]$ via $z_n\mapsto y(y-1)\cdots(y-n+1)$.

\entry{180508} Etingof: Mod $\epsilon^2$, co-Jacobi is not needed for $\delta$.

\entry{180507} {\bf Proj.} Unravel the topology behind Enqriquez-Furusho \arXiv{1605.02838} {\em A Stabilizer Interpretation of Double Shuffle Lie Algebras}; also at \href{http://scgp.stonybrook.edu/video/}{SCGP}.

\entry{180423} {\bf Riddle} (Masbaum). Can you partition a rectangle exactly one of whose sides is irrational into finitely many squares?

\entry{180413} Itai: $\{$partitions of $n$ into odd numbers$\}\leftrightarrow\{$partitions of $n$ into distinct numbers$\}$.

\entry{180411} {\bf Riddle} (Tsimerman). The area of a projection of a unit cube on a plane $P$ is the length of its projection on $P^\perp$.

\entry{180402} \bbs{VanDerVeen}{180402}{104238}: $y\leadsto x^{-1}$ via $y=(\omega+(T-T^{-1})a)x^{-1}$.

\entry{180320} $M_{2\times 2}(\bbC)/(\text{conj})$ isn't Hausdorff.

\entry{180318} In CU/QU, $t_i$'s are ``semi-scalars'' --- scalars for one tensor factor, Lie elements for $m$, $\Delta$, $S$. Reduction by $\langle t_i\rangle$ kills interest. $\langle t_i=t_j\rangle$ is interesting for $m$ but fully destructive for $\Delta$/$S$.

\entry{180317} Burton \arXiv{1712.05776}: HOMFLY-PT is $\bbe^{O(\sqrt{n}\log n)}$.

\entry{140405} In \href{http://drorbn.net/AcademicPensieve/Projects/Mathematica/nb/Localization.pdf}{Projects: Mathematica: Localization.pdf}: {\tt a=1; c:=b; Command[\{a=a, b=2\}, x:=a; y=c]; ?x; ?y}. \verb$Block$: local values, {\tt x:=a, y=2}. {\tt Module}: local symbols, {\tt x:=a\$1, y=b} with {\tt a\$1=1} and unset {\tt b}. {\tt With}: internal replacements, {\tt x:=1, y=b}.

\entry{170427} Faddeev \arXiv{math/9912078} (10), then Quesne \arXiv{math-ph/0305003}: $\log\bbe_q^x = \sum_{k\geq 1}\frac{(1-q)^kx^k}{k(1-q^k)}$. Readable proof in Zagier's ``\href{http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-30308-4_1/fulltext.pdf}{The Dilogarithm Function}'', pp.\ 28-30.

\entry{180206} ``Diagrammatic'' $\Rightarrow$ depends on $\frakg$ continuously; meaningful ``group-like''; an intrinsic upper bound on what can be done.

\entry{180208} The Benkart-Witherspoon representation of  \href{http://drorbn.net/AcademicPensieve/2017-06/nb/BW.pdf}{2017-06/BW.nb} is commented out here.
%Benkart-Witherspoon, \href{http://drorbn.net/AcademicPensieve/2017-06/nb/BW.pdf}{2017-06/BW.nb}: At $\gamma\epsilon\hbar=\sigma-\rho$, represented by
%$y\to\begin{pmatrix}0&0\\-\bbe^\rho&0\end{pmatrix}$,
%$a\to\frac{\gamma}{\rho-\sigma}\begin{pmatrix}\rho&0\\0&\sigma\end{pmatrix}$,
%$A\to\begin{pmatrix}\bbe^\rho&0\\0&\bbe^\sigma\end{pmatrix}$,
%$x\to\frac{\bbe^\rho-\bbe^\sigma}{\hbar \bbe^{\rho+\sigma}}\begin{pmatrix}0&1\\0&0\end{pmatrix}$,
%$t\to\frac{\rho+\sigma}{\hbar}\begin{pmatrix}1&0\\0&1\end{pmatrix}$,
%$T\to\bbe^{-(\rho+\sigma)/2}\begin{pmatrix}1&0\\0&1\end{pmatrix}$,
%$b\to\frac{\epsilon}{\sigma-\rho}\begin{pmatrix}\sigma&0\\0&\rho\end{pmatrix}$,
%$B\to\begin{pmatrix}\bbe^{-\sigma}&0\\0&\bbe^{-\rho}\end{pmatrix}$.

\entry{180205} {\bf Task. } Re-examine the relation between 2-parameter and 1-parameter quantum groups. Wherefore moding out by $t$? Can it be recovered? Where is $\epsilon$ hiding in the 1-parameter picture?

\entry{180126} Lua\TeX\ marries Lua and \TeX.

\entry{180118} Outreach talk idea ``Humans' Art and God's Art'': pattern recognition in Schwartz's factorization, then in an {\tt ArrayPlot} of primes, then a word on crypto, then K250 poster.

\entry{170520} $\uppertriang$ is graded! Related to spinners (sol's of $[a_{12},s_1+s_2]=0$ and $\delta\act [\cdot,\cdot]=[-,s]$). $s$ is inseparable from the tadpole $\text{\footnotesize$\overset{\circlearrowleft}{\bot}$} = (a_{12}-a_{21})\act m^{12}_-$ by products, co-products, primitivity, and degree. \refentry{180724}.

\entry{180113} Dylan: ``elevator pitch''.

\entry{180106b} Refined BCH: What's $\log\bbe^{x\to z}\bbe^{y\to z}$ in $\calA^{v,ac}(x^\ast y^\ast z)$? Is it of $z$-degree 1? What if replacing $\bbe^{?\to z}$ by another $\exp$?

\entry{180106a} In $\calA^{v,ac}$, can bring all $c$ vertices to before all $b$ vertices.

\entry{180104b} {\bf Q.} Let $B^U_L$ be the RAAG generated by $(R^a_x)_{a\in U,x\in L}$ modulo $(R^a_x,R^b_y)=1$ whenever locality, $a\neq b$ and $x\neq y$. Let $A^U_L=\gr B^U_L$, the RAAA generated by $(r^a_x)_{a\in U,x\in L}$ modulo $[r^a_x,r^b_y]=0$ whenever locality. Let $Z\colon B^U_L\to A^U_L$ by $R^a_x\mapsto\bbe^{r^a_x}$. Are there $\{\Delta_a^{bc},\Delta^x_{yz}\}$ on $A^U_L$ compatible with the natural ones on $B^U_L$?

\entry{180104a} Costello, Witten, Yamazaki \arXiv{1709.09993}: ``Gauge Theory and Integrability, I''.

\entry{171228b} Nosaka \arXiv{1712.02060}: ``the Orr invariant of degree $k$ is equivalent to the tree [] Kontsevich invariant of degree $<2k$''.

\entry{171228a} Gonz\`alez-Meneses, Silvero \arXiv{1712.01552}: Polynomial braid combing.

\entry{170829b} Przytycki \arXiv{1707.07733}: With HOMFLYPT $P(\bigcirc)=1$, $aP(\overcrossing)+a^{-1}P(\undercrossing)=zP(\smoothing)$, expand $P=\sum_iP_{2i}(a)z^{2i}$. Then $P_{2i}$ is of complexity $O(n^{2+3i})$, likely $O(n^{2+2i})$. {\bf Q.} Does $P_{2i}$ factor through type $2+2i$ invariants? Ito \arXiv{1710.09969}: (1) Related to ``low genus invariants'' in the $\calA\to\calM$ sense. (2) Even the cables of $P_0$ are mutation invariant.

\entry{171214} {\bf Wanted.} In $\calA^{\sim v}$, a tail-strand/head-strand pairing $P$, a co-product $P$-dual to $m$ and a $P$-compatible antipode ($R$ would then be the $P$-inverse of $P$?). Breaks: (1) A spinner / a homotopy $w$-strand. (2) The Cartan-criterion relation.

\entry{171213} {\bf Q.} What's transmutation? (In Majid and in Habiro's ``Bottom Tangles'').

\entry{171212} {\bf Q.} An $\calA^v$ analog of the $\calA^u$ notion ``$gl(N)$ genus 0''?

\entry{170923} {\bf Q.} Is $\hat\hbar$ injective on $\calU_\hbar(\frakg)$? What's $\gr\calU_\hbar(\frakg)$? Expansion? Is it inductive? Can I trust a non-universal inductivity proof?

\entry{171202} If $(C(S),\Delta^a_{bc})$ is a meta coalgebra (less is enough; ``symmetric set comodule?'') and $C^n\coloneqq C(\{0\}\cup\underline{n})$ then $d\colon C^n\to C^{n+1}$ by $dE\coloneqq\sum_{k=1}^{n+1}(-)^k E\act \sigma^{k,\ldots,n}_{k+1,\ldots,n+1} \act \Delta^0_{0k}$ has $d^2=0$. {\bf Q.} Find $H^n$ when $\Delta^t_{tx} f(t,S)$ is $D_1=f(t,S)$, $D_2=f(t+x,S)$ or $D_3=D_2-D_1$ on $\FA(\ast,S)$. For $D_3$, a spectral seq.\ with $D_{1,2}$?

\entry{170625} $\calU_{\hbar;\gamma\epsilon}$ conventions in \href{http://drorbn.net/AcademicPensieve/Projects/PPSA/CS-PPSA.pdf}{Projects: PPSA: CS-PPSA.pdf}.

\entry{171117} Kotorii \arXiv{1705.10490}: $n$-equivalence on $\vK(\uparrow)$ $\Leftrightarrow$ equivalence modulo $\operatorname{LCS}(\PvB)$.

\entry{171116} The proofreader's \raisebox{-5pt}{\input{figs/TransposeClasper.pdf_t}}.

\entry{171109} Cheng, Jackson, Stanley \arXiv{1601.01377}: With $q = \bbe^{\hbar/2}$, $(n)_q = (q^n-q^{-n})/(q-q^{-1})$ under $[h,x] = 2x$, $[h,y] = -2y$, $[x,y] = (h)_q$, have $\frac{x^a}{(a)_q!} \frac{y^b}{(b)_q!} = \sum_{i\geq 0} \binom{h+b-a}{i}_q \frac{y^{b-i}}{(b-i)_q!} \frac{x^{a-i}}{(a-i)_q!}$ (and more). Also in \href{http://drorbn.net/AcademicPensieve/People/VanDerVeen/nb/Generalxy.pdf}{People: VanDerVeen: Generalxy.nb}.

\entry{171108b} {\bf Def.} $\calK=\calK^{\text{\it gcs-rvt}}$: ground (some components are on the ground) ceiling (some are ceiling) surgery (some $g/c$ components are surgery-slippery) rotational virtual tangles. $\calK$ is $\OU$.
{\bf Conj.} (1) $\calA^{\text{\it gcs-rvt}}$ has a combinatorial description. (2) $\calK$ has a surgery-compatible expansion; v-Hopf surgeries split. (3) $\calK$ has a stitching-compatible ``universal dequantizator'' expansion.

\entry{171108a} {\bf Q.} Do volumes / homologies extend to rotational v-knots?

\entry{171018} Taylor: If $f(a+x)=\sum_{k=0}^n\frac{f^{(k)}(a)x^k}{k!} + R_{n,a}(x)$ then $\exists\xi_{1,2}\in(0,x)$ s.t.\ $R_{n,a}(x)=\int_0^x\frac{f^{(n+1)}(a+\xi)}{n!}(x-\xi)^nd\xi = \frac{f^{(n+1)}(a+\xi_1)}{n!}x{(x-\xi_1)^n} = \frac{f^{(n+1)}(a+\xi_2)}{(n+1)!}x^{n+1}$.

\entry{171107} Manin '89 ``multiparametric quantum deformation''. Garcia, Gavarini \arXiv{1708.05760} ``multiparameter quantum groups'' (MpQG).

\entry{140213} \$: $\digamma$, {\tt mathtools:} $\coloneqq$, {\tt mathabx:} $\lefttorightarrow\Neptune$, {\tt babel:} {\greektext\sampi\qoppa}, {\tt txfonts:} $\multimapdotbothvert$. Hupfer \href{http://tex.stackexchange.com/questions/219839/jumplines-continued-on-page-in-latex}{knows} jumplines. \pdftooltip{{\red pdfcomment}}{pdfcomment has plenty popups, tooltips, margin notes, etc.}.

\entry{171102} Khovanskii: an algebraic formula is expressible in radicals iff its monodromy group is solvable. $\Rightarrow$: Arnol'd argument. $\Leftarrow$ middle lemma: a finite Abelian group $A$ acts on a ring $R$ that contains all roots of 1. Then every element of $R$ is a linear combination of roots of elements in $R^A$.

\entry{171015} Ito's \arXiv{1411.5418} ``Topological formula of the loop expansion of the colored Jones polynomial'' has (multi-)forks.

\entry{171010} An expansion $\calU_{\hbar;\gamma\epsilon}\to\gr_\epsilon\calU_{\hbar;\gamma\epsilon}$?

\entry{171009b} {\bf Q.} A name for $\bbe^{\nu(\xi x+\eta y+\delta x y-t\xi\eta)}$?

\entry{171009a} {\bf Q.} If $\phi\colon(V=\bbR^n_{\xi^i})\to(W=\bbR^m_{\eta^j})$ and $W=\langle y_j\rangle$ with $\eta^j(y_k)=\delta^j_k$, what do you call $\Phi(\xi^i,y_j)\coloneqq\sum y_j\phi^\ast(\eta^j)$?

\entry{141226b} {\bf Proj.} FT invariants of fixed-linking-numbers (uvw)-KO. A Goussarov view? FT relative to CO/CU (Commute Overcrossings/Undercrossings)? Is there a good presentation of tangles with fixed linking numbers?

\entry{170919} Talk idea: ``$\OU$-Tangles \& the Quantum Groups Conspiracy''.

\entry{170128} Cartan's criterion: $\frakg\subset\End(V)$ is solvable iff $\forall x\in\frakg,\,y\in[\frakg,\frakg],\ \tr_V(xy)=0$. Induces a quotient of $\calA^v$; what is it?

\entry{170917} \href{https://people.math.osu.edu/gautam.42/}{Gautam}: an explicit $\calU(sl_n)\llbracket\hbar\rrbracket \cong \calU_\hbar(sl_n)$.

\entry{170914} {\bf Q.} Is there a canonical isomorphism between quantizations of $sl_2$ with varying $r$?

\entry{170913b} {\bf Do.} Center poly-poly at Lie algebra contractions.

\entry{170908} Livingston \arXiv{1709.00732}: $\sigma\colon S^1\to\bbZ$ is a knot signature function iff all discontinuities are at roots of an Alexander polynomial and $[\ldots]$. {\bf Q.} How fits with w and with $\Gamma$-calculus?

\entry{170519} Bonahon's ``miraculous cancellations'' \arXiv{1708.07617} link Ito with PPSA?

\entry{170813} Given a Hopf $H$, is there a ``pair one'' op $H^\ast\otimes H\to H^\ast\otimes H$?

\entry{170309} In \href{http://drorbn.net/AcademicPensieve/2017-03/nb/geps.pdf}{2017-03/geps.nb}: $w,u,b,c=$ $\begin{pmatrix}0&1\\0&0\end{pmatrix}$, $\begin{pmatrix}0&0\\-\epsilon&0\end{pmatrix}$, $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$, $\frac12\begin{pmatrix}-1-\epsilon^{-1}&0\\0&1-\epsilon^{-1}\end{pmatrix}$ obey $\frakg^\epsilon$: $[w,c]=w$, $[c,u]=u$, $[u,w]=b-2\epsilon c$. Then $r=(b_1-\epsilon c_1)c_2+u_1w_2=\frac{1}{4\epsilon}\begin{pmatrix}1-\epsilon&0\\0&1+\epsilon\end{pmatrix} \otimes \begin{pmatrix}1+\epsilon&0\\0&1-\epsilon\end{pmatrix} - \epsilon\begin{pmatrix}0&0\\1&0\end{pmatrix} \otimes \begin{pmatrix}0&1\\0&0\end{pmatrix}$.

\entry{170807} Naef: For $D\in\der(\FL)$ (any!), $\atdiv(D)\coloneqq\sum_x\tr\partial_xD(x)$ satisfies $\atdiv[D_1,D_2]=D_1\atdiv(D_2)-D_2\atdiv(D_1)$.

\entry{170804} Given $f(x)$, $g(\xi)$, have $f(\partial_\xi)g(\xi)|_{\xi=0}=g(\partial_x)f(x)|_{x=0}$.

\entry{170802} Does every infinitesimal deformation of a solvable Lie algebra globalize? Does some $H^2$ parameterise knot invariants?

\entry{170708} In w, inner $q\Delta$'s automatically lead to $\sder$- hence tree-level u- associators.

\vskip -1mm
\needspace{7mm} % 6mm is too little.
\parpic[r]{$\xymatrix@C=12pt@R=12pt{
  \calA^w(\uparrow) \ar[r]^? \ar[d]^\calT &
    \calA^w(\uparrow_2) \ar[d]^\calT \\
  \calU(sl_2^0) \ar[r]^{q\Delta} &
    \calU(sl_2^0)^{\otimes 2}
}$}
\entry{170707} Which doubling makes the diagram commute?

\entry{170703} What characterizes ``PBW'' maps $\calS(\frakg)\to\calU(\frakg)$?

\entry{170702} In $\calU_{\hbar;\gamma\beta}$, $\prod_i \bbe^{\eta_iy}\bbe^{\alpha_ia}\bbe^{\xi_i x} = \bbe^{\eta y}\bbe^{\alpha a}\bbe^{\xi x}\bbe^\sigma\left(1+\sum_{k\geq 1}\Lambda_k\beta^k\right)$, with $\alpha = \sum\alpha_i$, $\eta = \sum_i\eta_i \bbe^{-\gamma\sum_{j<i}\alpha_i}$, $\xi = \sum_i\xi_i\bbe^{-\gamma\sum_{j>i}\alpha_i}$, $\sigma = \frac{1-T}{\hbar}\sum_{i<j}\xi_i\eta_j\bbe^{-\gamma\sum_{l\colon i<l<j}\alpha_l}$ and $\Lambda_k$ is \ldots

\entry{170412b} Rote (2001), \bbs{Dancso}{170529}{141410}: An $n^4$-time division-free algorithm for $\det$.

\vskip -1.5mm\needspace{0mm}
\parpic[r]{\scalebox{0.85}{\input{figs/PairingPlumbing.pdf_t}}}
\picskip{3}
\entry{170602} Word-pairing in Hopf algebras:
\[ \left\langle \prod_{i\in \underline{n}} x_i,\, \prod_{j\in\underline{m}} y_j \right\rangle =
  \prod_{i\in \underline{n},\,j\in\underline{m}} \left\langle x_i^{(j)},\, y_j^{(i)} \right\rangle.
\]

\entry{170529} Is ``$(\frakg,[],\delta)$ non-negatively graded with Abelian degree 0'' same as ``$D\frakg$ is a sum of Kac-Moody and inhomogeneous factors''? A sense by which these are precisely ``the quantizeables''?

\entry{170323b} Is there a ``Heisenberg-Drinfel'd Double Construction''?

\entry{170224} $\frakg$ solvable $\Leftrightarrow$ $[\frakg,\frakg]$ nilpotent $\overset{?}{\Leftrightarrow}$ $\frakg \cong \fraka\ltimes\frakn$ with Abelian $\fraka$ and nilpotent $\frakn$. Also, $\frakg$ solvable $\Leftrightarrow$ there is a finite decreasing filtration $(\frakg_k)_{k\geq 0}$, $\frakg_0=\frakg$, with $\frakg_0/\frakg_1$ Abelian and $[\frakg_k,\frakg_l]\subset\frakg_{k+l}$. Then $\calU(\frakg)$ also has a multiplicative decreasing filtration.

\entry{131009} Let $\Gamma_{1,2,3}$ be thickened surfaces. Is there an expansion for the structure $\calK(\Gamma_1\hookrightarrow\Gamma_2)\times\calK(\Gamma_2\hookrightarrow\Gamma_3) \overset{\sslash}{\rightarrow} \calK(\Gamma_1\hookrightarrow\Gamma_3)$? (Handlebodies in Habiro-Massuyeau \arXiv{1702.00830}).

\entry{170518} Given unital algebras $B$, $C$ and $R\in B\otimes C$, when is there a swap $s\colon C\otimes B\to B\otimes C$ so that $R^{12}R^{13}R^{23} = R^{23}R^{13}R^{12}$ would hold in $B\bowtie_sC$?

\entry{170512} Le: ``Every $\calU_q(\frakg)$ embeds in some quantum torus''. A smidge version? Does ''every $\frakg$ embeds in a Heisenberg'' quantize to Le's?

\entry{170508} Majid's Primer \S 4: $A\mapsto M(A)$, algebras to bialgebras.

\entry{131023} Markl: ``like a bottle under a waterfall''. Psychology buzz\-word: ``cognitive overload''.

\entry{170413a} Shortcut / characterize / replace / generalize $\calU(\frakg)^\ast \cong \calS(\frakg)^\ast \cong \calS(\frakg^\ast) \cong \calU(\frakg^\ast)$. Are ``doubling the Cartan'' a/o ``$\Delta$-conjugations'' more fundamental than ``pairing''?

\entry{131130b} {\bf Proj.} ``Alexander Recovery''. Conway relation; relations as in Archibald; factorization as in Levine \arXiv{q-alg/9711007}, Tsukamoto-Yasuhara \arXiv{math/0405481}; cabling; Fox-Milnor (is there for links?); genus property; crossing-number property; split-link property; u-range; w-range; unitarity; concordance; mutations; behaviour under mirror/strand reversal; Torres conditions; Hartley's property; cheirality properties as in \arXiv{1608.04453}; Alexander/Thurston norms as in \href{http://abel.math.harvard.edu/%7Ectm/papers/home/text/papers/alex/alex.pdf}{McMullen:\linebreak[0]alex.pdf}.

\vskip -2mm\needspace{8mm} % 7mm is not enough.
\parpic[r]{$\xymatrix@C=5mm@R=5mm{
  \calA^v(\uparrow) \ar[r]^{T_{\frakg}} \ar[d]^\Delta &
    \calU(\frakg) \ar@{<->}[d] \\
  \calA^v(\uparrow\uparrow^{ab}) \ar[r]^<>(0.5){T_{\frakg^+}} &
    \calU(\frakg^+)
}$}
\entry{170413b} In \href{http://drorbn.net/AcademicPensieve/2009-01/KAL-090128-_Lie_bialgebras,_gl(N),_framing_v-knots.pdf}{2009-01/KAL-090128$\ldots$.pdf} and \bbs{KAL}{090128}{160808}:

\entry{170411} \href{https://www.msri.org/workshops/826}{Vogtmann@MSRI}: action of trivalent trees on the $\gr(\pi_1(\Sigma_g))$ by derivations via contractions; relation between $H^\ast(\Out(F_n))$ and some $H^\ast$ of trees modulo AS \& IHX.

\needspace{0mm}
\entry{170211b} Gaussian pairing:
$\left\langle
  \exp\left(\frac{x\subset}{2}\right)
  \mid
  \exp\left(\frac{\supset y}{2}+\sum_{i}\multimapdot\!i\right)
\right\rangle =$
\newline\null\hfill$\exp\left(\frac12\log(\frac{1}{1-xy})\bigcirc
  + \sum_{i,j}\frac{x\,i\multimapdotboth j}{1-xy}
\right)$.

\entry{131014} Problems with the projectivization paradigm: No room for negative degrees and for degree-decreasing ops. No built-in $\hat{\hbar}$.

\entry{170325b} {\bf Q.} Is there a ``Vogel group'' acting on $\calA^w(S)$? In general, is there topology behind the Vogel action?

\entry{170325a} Generalized Weyl: If $f\in\calS(V)$ and $\psi\in\calS(V^\ast)$ then in $\calU(HV)$, $f\psi = \psi_1f_1\langle\psi_2,f_2\rangle$, where $\Delta f=\sum f_1\otimes f_2$ and $\Delta\psi=\sum\psi_1\otimes\psi_2$. Is there a version with $\calU(\frakg)$ replacing $\calS(V)$?

\entry{170325c} {\bf Q.} Is there a $\calK^u$ interpretation of the Vogel action on $\calA^u$?

\entry{170318} {\bf Q.} Are there easy $\theta$-invariant braidors for $\frakg_0$?

\entry{170323a} {\bf Proj.} Study $\calK^w/\calA^w$ with ``solvable heads''.

\entry{170322} With $f_t=\bbe^t-1$, $f_{x+y}=f_x+\bbe^xf_y=\bbe^yf_x+f_y$.

\entry{170320a} {\bf Q.} Is there a good algebraic structure of ``groups with a fixed Abelianization''?

\entry{170310} Tentative ID: I'm a selfish rationalist atheist permissive individual-rights free-market socialist global citizen. Note to self: read defs! (Liberal? Democrat?)

\entry{170308} Mine ``$\calS(\frakg)\colon\calS(\frakg^\ast)$ pairing'' $\Leftrightarrow$ ``commutation in $\calU(H\frakg)$''.

\entry{170302} {\bf Conj.} Every rotational classical YB structure can be quantized. Related to spectral parameters? (Chari-Pressley 15.2.B).

\entry{170306} For $sl_2^+=\langle e,f,h,c\rangle/([h,e]=2e,\,[h,f]=-2f,\,[e,f]=h,\,[c,\cdot]=0)$, $r_{ij}\coloneqq e_if_j+h_ih_j/4+\alpha(h_ic_j-c_ih_j)$ solves CYBE.

\entry{170301} {\bf Prob.} Given $\ad$ and a solvable structure on $\frakg$, fully implement the group-likes in $\hat\calU(\frakg)$.

\entry{170223d} VdV on gmail/161122: A $\frakg_0$ / $gl(1|1)$ relationship.

\entry{170223c} {\bf Q.} What's the internal kernel in $\calA^v$ for 2-loop $Z^u$? What's the nearest-dual Lie bialgebra?

\entry{170223b} {\bf Do.} Compute $Z_{I\frakg}$ for general $\frakg$ w/o back reference to $\calA^w$.

\entry{170223a} By nilpotent approximation, all semi-simple weight systems come from nilpotent Lie algebras. Do the latter make more?

\entry{170222} Is there a Chern-Simons theory for degenerate Casimirs?

\entry{141107} Claim. $\frakg$ a Lie algebra, $d\in\frakg$ fixed, $c$ a ``new'' central element, $\frakg_1\coloneqq\frakg\oplus\langle c\rangle$, $\delta\colon\frakg_1\to\frakg_1\otimes\frakg_1$ by $c\mapsto 0$ and $\frakg\ni x\mapsto[d,x]\otimes c+c\otimes[d,x]$, then $\frakg_1$ is a Lie bialgebra. Extends to a non-cocommutative seed? Eckhard: may be related to Medina-Revoy ``double extensions'', a structure theorem for metrized Lie algebras. \refentry{141114a}.

\entry{170221} Teichner on \mo{7052}: $K$ is slice iff $K\#R$ is ribbon for some ribbon $R$.

\entry{170126b} {\bf Q.} What does Ado give for $\frakg_1$? For $I\frakg$ (\refentry{190709})?

\entry{161027a} Describe $\calB^{rv}$ and $\alpha\colon\calA^u\to\calA^{rv}\coloneqq\calA^v/\langle[a_{ij},s_i+s_j]\rangle$.

\entry{170126a} From Roland's {\tt Poly.pdf}: Under $[F,E]=1-t-(1+t)\epsilon L$, $[F,L]=F$, $[L,E]=E$, $t=e^c$, $s=1-t$ and $\nu=(1-s\delta)^{-1}$, have
$\displaystyle \bbO(FE|e^{\alpha E+ \beta F +\delta EF+\epsilon (s-2)P(E,F)}) =$
\newline$
  \bbO\big(ELF|(1+\epsilon(s-2)(P(\partial_\alpha,\partial_\beta) + \partial_s((\partial_\alpha+\partial_\beta)/2+\partial_\delta+L)+s\partial^2_s/4))
$
\newline\null\hfill$
  \nu e^{\nu(\alpha\beta s + \alpha E+ \beta F+ \delta EF)}\big)
$

\entry{131103} The Yoshikawa moves (usefulness limited by the before/after unknottedness condition; Chterental: the Swenton proof of completeness may be broken, new ones by Kearton-Kurlin and himself exist):
\[ \includegraphics[width=\columnwidth]{figs/YoshikawaMoves.png} \]

\entry{170108a} {\tt Table[As[n,6], As[k,0,n], Binomial[n,k]]} in \href{http://drorbn.net/AcademicPensieve/2017-01/nb/As.pdf}{2017-01/As.nb}.

\entry{170107b} What do coverings of the annulus say about annular braids?

\entry{170107a} Is there strand-doubling for handle-strands ($|_h$'s)? In general, what do maps between surfaces of (possibly different) genera say about $\calA(|_h^g\uparrow^n)$?

\entry{161122} What's $\theta$, in $\calA^w$ language?

\entry{161101} Stein's paradox: with $\theta\in\bbR^{n\geq 3}$, given a single measurement $x_i$ of $n$ independent normal Gaussians with mean $\theta_i$ and variance $1$, the estimator $\hat\theta=x$ for $\theta$ is dominated by another (across all $\theta$), if aiming to minimize $E\left[\left\|\theta-\hat\theta\right\|^2\right]$.

\entry{161027b} What's the abstract relation between Roland's $\frakg_0$ and mine?

\entry{161009} Kondo (1979): Any Alexander polynomial is attained with unknotting number 1.

\vskip -1.5mm \needspace{11mm} % 10mm is not enough.
\parpic[r]{\input{figs/ScottsEquation.pdf_t}}
\entry{160911} {\bf T/F?} The only solutions to Morrison's equation for $p\in\PB_4$ are $1$ and $\sigma_2^{-2}$.

\entry{160801} Is there a topological meaning to primitive-exponential hybrids like $\Phi^{-1}t_{14}\Phi$?

\entry{160721} {\bf Q.} Are braidors related to quandle cohomology?

\entry{160616} Representing $\calA^w$ on functions on a 2D Lie algebra, what is the functional representation of the braid group we get?

\entry{160613} Find Gassner and dual-Gassner in the topology of $\PwB_n$.

\entry{160609} $\rad\frakg\coloneqq$(maximal solvable ideal). Levi: $\rad\frakg$ has a complementary Lie algebra.

\entry{160519} {\bf Proj.} Extendibles extend to extendibles, the group case.

\vskip -1.5mm\needspace{0mm}
\parpic[r]{\input{figs/MrowkasDodecahedron.latex}}
\entry{160508} Mrowka's dodecahedron:

\entry{160505} Chterental: Brunnian 2-component links have manifestly Brunnian diagrams.

\entry{160503b} {\bf Prob.} Find a methodology for promoting invariants of braid-like virtuals to full invariants of classicals.

\entry{160503a} {\bf Q.} What's $\omega$, as seen from linear control theory?

\entry{160414b} {\bf Q.} In $\bbQ G$, is $\varprojlim_{m\geq n}I^n/I^m=\left(\varprojlim_mI/I^m\right)^n$?

\entry{160414a} {\bf Q.} Suppose $A\otimes B\surjects C$ and all are filtered compatibly. Does $\widehat{A\otimes B}\surjects \hat{C}$? Does $\hat{A}\otimes\hat{B}\surjects\hat{C}$?

\entry{160410} {\bf Q.} Can the filtrations of $\bbQ G^{(n)}$ and of $\widehat{\bbQ G}$ be defined from their (Hopf-)algebraic structures?

\entry{160408} {\bf Q.} Integrate Lie algebra 2-cocycles to Lie group 2-cocycles.

\entry{160403b} {\bf Prob.} Characterize $\left\{\exp F(\log X,\log Y)\right\}$ in $\widehat{\bbQ\FG_2}$ language.

\entry{160330} {\bf Proj.} An ``Insolubility of the Quintic'' web site.

\entry{160315b} Is $\FG_{a,b,c,d}/\{a=c^b,b=d^a,c=e^f,d=f^e\}$ free? Expansion faithful?

\entry{160315a} $\pi_2(\text{$n$-ring complement})=\bbZ(\FG_n\times\underline{n})$?

\entry{160311} For $\wB$, why does the $\pi_1$ action determine the $\pi_2$ action? For $\wT$, it doesn't.

\entry{160308} Naor's $\sqrt{2}\notin\bbQ$: Else $0<(\sqrt{2}-1)^n\to 0$ but with $\sqrt{2}=p/q$, $(\sqrt{2}-1)^n=a_n\sqrt{2}+b_n = (pa_n+qb_n)/q \geq 1/q$.

\entry{160304} {\bf Riddle} (saw $\Omega$ by Gracia-Saz after M.~Bernstein). 100 prisoners strategize, then are sealed in rooms with the same countable sequence of ``boxes with reals'' in each. Can each open all but one of their boxes and guess the remaining one so that at most one prisoner would be wrong? Hint: 0-1 boxes, finitely many 1s.

\entry{160113} A meta-monoid $M$ is {\em factored} if it has a $\Box$ compatible with all operations. What structure form the primitives $P$ of $M$? Does $P$ determine $M$?

\entry{151214} \bbs{Schneps}{151209}{100312}, the spherical $\pentagon$:
\[ \varphi(t_{12},t_{23}) \varphi(t_{34},t_{45}) \varphi(t_{51},t_{12}) \varphi(t_{23},t_{34}) \varphi(t_{45},t_{51}) = 1. \]

\entry{151209} Are there solutions of R4 (+more?) in $\langle a_{12},a_{21}\rangle$?\hfill No?

\entry{150924} Schneps in Les Diablerets: For $f\in\FL(x,y)$, $\pi_y(f)$ proj.\ on words ending with $y$, $f_\ast\coloneqq\pi_y(f)-\sum\frac{(-)^n}{n}(f\mid x^{n-1}y)y^n$ rewritten in $y_i\coloneqq x^{i-1}y$. $\partial s\coloneqq\{f\colon\Delta_\ast(f_\ast)=f_\ast\otimes 1+1\otimes f_\ast\}$, with $\Delta_\ast(y_i)\coloneqq\sum_{k+l=i}y_k\otimes y_l$ (group version in \bbs{Schneps}{151209}{123837}). \bbs{Ens}{150923}{183209}: Write $u,v\in\FA(x,y)y$ in $y_i\coloneqq x^{i-1}y$ and set $\St(1,u)=\St(u,1)=1$, $\St(y_iu,y_jv) = y_i\St(u,y_jv)+y_j\St(y_iu,v)+y_{i+j}\St(u,v)$. Then $\partial s=\{f\in\FL_{\geq 3}(x,y)\colon(f\mid\St(u,v))=0\}$, where not both $u$ and $v$ are powers of $y$. For $f\in\partial s$ set $F(x,y)=f(-x-y,-y)=xF^x+yF^y$, $G(x,y)=\sum_{i\geq 0}\frac{(-)^i}{i!}\partial_x^i(F^x)yx^i$. Then $f\mapsto D_{F,G}$ is $\partial s\hookrightarrow\krv_2$.

\entry{151003a} \bbs{Ens}{151002}{144503}: In $\FA(u_d)$ with $\deg u_d=d$, if $\exp\sum u_d=\sum y_k$ with $\deg y_k=k$, then $\Delta y_k=\sum_{i+j=k}y_i\otimes y_j$.

\entry{151126} $A$ a Hopf algebra, $b$ a primitive derivation $b\act\Box = \Box\act(b\otimes 1 + 1\otimes b)$, $B\coloneqq \{D\in A\colon (bD=0)\wedge(\Box D=D\otimes 1 + 1\otimes D)\}$. Characterize the subalgebra $\langle B\rangle$ generated by $B$.

\entry{151118c} Wikipedia: Schur multiplier: ``A projective representation of $G$ can be pulled back to a linear representation of a central extension $C$ of $G$''.

\entry{151118b} Hopf: $F$ free, $G=F/R$,\hfill$H_2(G,\bbZ)\cong(R\cap[F,F])/[F,R]$.

\entry{151118a} Hillman's Alg.\ Inv.\ of Links, pp. 238: ``Cochran, Orr conjectured that if all Milnor invariants of length $<r$ vanish then all to length $2r$ are well-defined''.

\entry{151110} The Milnor homotopy trick $[x_i,x_i^g]=1$ by Habegger-Lin:
\newline\includegraphics[width=\linewidth]{figs/MilnorHabeggerLinHomotopyTrick.png}

\entry{151012} The Goldman Lie algebra, which is its group?

\vskip -2mm
\needspace{17mm} % 16mm is not enough.
\parpic[r]{\includegraphics[width=2.25in]{figs/LD15_Kuno-1.png}}
\entry{151006} Kuno@LD15: (missing: the archetypical model for ``$\sigma$ is an isomorphism'')

\entry{151003b} The Alexander quandle: $t\colon A\to A$ an automorphism of an Abelian group, $a\uparrow b\coloneqq ta+(1-t)b$.

\entry{131122c} Overhand/underbelly, \hspace{-2mm}$\begin{array}{c}\includegraphics[width=1.65in]{figs/OverhandUnderbelly.pdf}\end{array}$\hspace{-1mm}, abstain. Sep.\ 2015, \sout{stronger: ``4-end bottom tangles are unparenthesized''!}

\entry{150925} K\"onig's lemma: in an infinite connected graph with finite valencies there's an infinite simple path.

\entry{140123}
$\geq 47$ 4D hardware pieces at \href{http://drorbn.net/AcademicPensieve/2013-12/4DHardware/}{2013-12/4DHardware/}:
{\def\randpiece{\hfill
\RandomZ{\kk}{1}{47}\ifnum \kk<10
  \includegraphics[height=0.19\columnwidth,width=0.19\columnwidth,keepaspectratio=true]{../../2013-12/4DHardware/4DHardware-0\kk.png}
  \else\includegraphics[height=0.19\columnwidth,width=0.19\columnwidth,keepaspectratio=true]{../../2013-12/4DHardware/4DHardware-\kk.png}
\fi}
\newline\randpiece\randpiece\randpiece\randpiece\randpiece}

\entry{131106a} XII $\Leftrightarrow$ FiC $\Leftrightarrow$ ``bra-cobra'' $\Leftrightarrow$ ``involutive'' (Chas, \arXiv{math/0105178}) $\Leftrightarrow$ ``infinitesimal of $S^2=1$''. \v{S}evera in 2015 Les Diablerets talk: this globalizes.

\entry{150829} Massuyeau: Passi, Passman: over $\bbQ$, dimension$=$LCS.

\entry{150806} Two $1/3$ rotations $\rho_{3|3'}$ on $\calA^w(\uparrow_x\uparrow_y)$: $S^y \act \Delta^y_{yz} \act m^{xz|zx}_x \act \sigma^{xy}_{yx}$. In general, $\Aut(\FG_n)\actsonleft\calA^v(\uparrow^n)$ and $\Out(\FG_n)\actsonleft\calA^u(\uparrow^n)$.

\entry{150804} {\bf TIL.} {\tt ctrl-alt-F1} through {\tt ctral-alt-F7}, {\tt pstree}.

\entry{150729} Habiro ring: $\widehat{\bbZ[q]}\coloneqq\varprojlim\bbZ[q]/(q)_n$ w/ $(q)_n\coloneqq \prod_{i=1}^n(1-q^i)$.

\entry{150719b} Odd quandle-from-group: $a\uparrow b\coloneqq ba^{-1}b$.

\entry{150719a} Gordon on Wada: With $\pi(K)\coloneqq  \smash{\raisebox{-3mm}{$\tensor*[_x^z]{\text{\huge$\overcrossing$}}{_y^x}$}}\to x^{-1}zx^{-1}y=1$, $\pi(K)\cong\pi_1(\Sigma_2(K))\ast\bbZ$.

\entry{150205} Abe-Tagami \arXiv{1502.01102}, Gompf-Scharlemann-Thompson \arXiv{1103.1601}: slice-ribbon counterexamples?

\entry{150624} ``Set function $\varphi\colon G\to H$ is affine'' means $\varphi(I^n_G)\subset I^n_H$. Makes a category. Group morphisms and translations are affine. $\varphi_i\colon G_i\to H$ affine $\Rightarrow$ $\varphi_1\varphi_2$ affine, so ``sorting'' on $\FG$ is affine, $Id\colon G\times H\to G\ltimes H$ is affine.  If $G\ltimes H$ is almost-direct, $Id\colon G\ltimes H\to G\times H$ is affine, so combing braids is bi-affine.

\entry{150517} $G\eqqcolon G_1$ a group, $G_{n+1}\coloneqq(G,G_n)$, $\pi_n\colon G_n\to L_n\coloneqq G_n/G_{n+1}$. Given affine sections $\varphi_n\colon L_n\to G_n$ let $\zeta\colon G\to \hat{L}\coloneqq\prod L_n$, the ``LCS-expansion using $\varphi_\cdot$'' (``group-PBW for $\varphi_\ast$''?), by $\zeta_1\coloneqq\pi_1$ and $\zeta_n(g)\coloneqq\pi_n(\varphi\zeta_{<n}(g)^{-1}g)$ where $\varphi(\lambda_1,\lambda_2,\ldots) \coloneqq \varphi_1(\lambda_1)\varphi_2(\lambda_2)\cdots$. Then $\zeta_{<n}(h)=0$ iff $h\in G_n$ and $g=\varphi\zeta_{<n}(g)$ in $G/G_n$. Is $\zeta_n$ of type $n$?

\entry{150522} Automatic structure on a group $G$: $A$ a set of semigroup generators, ``acceptor'' automaton $M$ on $A$ accepts $L$ s.t.\ $\pi\colon L\twoheadrightarrow G$, for each $x\in A\cup\{e\}$ ``multiplier'' automaton $M_x$ on $(A,A)=(A\times\$)\times(A\times\$)\setminus(\$,\$)$ accepting $(u_1,u_2)$ iff $\exists v_i\in L$ with $u_i\in v_i\$\ast$ and $\pi(v_2)=\pi(v_1x)$.

\entry{150609c} Budney's \arXiv{math/0309427}: Long knot space is $\bigsqcup_{n=0}^\infty(\calC_2(n)\times\calP^n)/S_n$, with $\calC_2(n)$ the space of $n$ little 2-cubes and $\calP$ the set of prime knots, with $[0,1]^2_{t,s}$ parameterizing arc length ($t)$ and scale ($s$).

\entry{150609b} \bbs{Lambrechts}{150603}{080914}: $H_\ast(\Emb(\bbR,\bbR^{\geq 4}))\cong H_\ast($a graph complex$)$. \bbs{Lambrechts}{150603}{080919} Goodwillie-Sinha: At $n\!\geq\!4$, $\operatorname{holim}_{p\to\infty}\widebar{\text{\it Conf}}^f(\underbar{p},\bbR^n) \!\simeq\! \Emb^f(\bbR,\bbR^n)$. Naive? Intuition?

\entry{150609a} Is $\pi_{>0}$ ever useful to understand $\pi_0$?

\vskip -2mm
\needspace{2mm} % 1mm is not enough.
\parpic[r]{\includegraphics[height=0.6in]{../../2015-06/vK21Pogs.png}}
\entry{150608} A {\tt PogForm} in \href{http://drorbn.net/AcademicPensieve/2015-06/}{2015-06}:

\entry{150502} \href{http://drorbn.net/bbs/show.php?prefix=Martins}{BBS:Martins}: $\bullet$ A Crossed Module (CM, e.g.\ \arXiv{0801.3921}) models $\partial\colon\pi_2(X,A)\to\pi_1(A)$: a group homomorphism $\partial\colon E\to G$ with an action $\triangleright\colon G\lefttorightarrow E$ s.t.\ (1) $\partial(g\triangleright e)=g(\partial e)g^{-1}$, (2) $(\partial e)\triangleright f=efe^{-1}$ (contains $\pi_1\coloneqq\coker\partial$, $\pi_2\coloneqq\ker\partial$, and the Postnikov $k$-invariant in $H^3(\pi_1,\pi_2)$ when $A=X^1$; equivalent to a ``2-group''). There are homotopies of CMs, the free CM over a set-to-group $\partial_0\colon C\to G$, quotients of CMs, the ``actor'' CM $G\to\Aut(G)$. Whitehead (JHC): $\pi_2(X^2,X^1)$ is the free CM over the attaching maps.

$\bullet$ A Differential Crossed Module (DCM, Baez-Crans \arXiv{math/0307263}, Cirio-Martins \arXiv{1309.4070}) is a Lie algebra morphism $\partial\colon\frakh\to\frakg$ with an action $\triangleright\colon\frakg\lefttorightarrow\frakh$ by derivations, s.t.\ (1) $\partial(g\triangleright h)=[g,\partial h]$, (2) $(\partial h)\triangleright h'=[h,h']$. Assign CM to DCM by $G\coloneqq\{e^\gamma\colon\gamma\in\frakg\}$, $H\coloneqq\{e^\eta\colon\gamma\in\frakh\}$, $\partial\colon e^\eta\mapsto e^{\partial\eta}$, $e^\gamma\triangleright e^\eta\coloneqq e^{e^{\gamma\triangleright}\eta}$. There's an analytic $\{\text{\it CM}\}\to\{\text{\it DCM}\}$; not yet algebraic. Use $\Rker$, $s,t\colon E\rtimes G\to G$ (\bbs{Martins}{150501}{150100})?

$\bullet$ There's a DCM $\calG\calL(\calV) = (\frakg\frakl_1(\calV) \to \frakg\frakl_0(\calV))$ for a chain complex $\calV$.

$\bullet$ Braided surfaces have $(\smoothing\to\slashoverback)$ (see Khovanov-Thomas \arXiv{math/0609335}).

$\bullet$ \bbs{Martins}{150501}{114927}: A CM $\pi_{12}(K^c)$ for virtual 2-knots.

$\bullet$ \bbs{Martins}{150501}{150100}: $\Rker$ for Hopf morphisms.

\entry{150417} \LaTeX\ displayed equations: {\tt equation(*)}, {\tt align(*)} \verb$BT(&T)*\\...E$), {\tt multline(*)}, {\tt split} (inner for displayed, \verb$BT&T\\...E$), {\tt aligned} (inner {\tt align}). Related: \verb$\label$, \verb$\tag$, \verb$\nonumber$, \verb$\notag$. [\pdftooltip{{\red WB}}{Wikibook: LaTeX / Advanced Mathematics}\href{http://en.wikibooks.org/wiki/LaTeX/Advanced_Mathematics}{$\to$}].

\entry{150422} Lambert's dreaded $W$ function: $y=xe^x\Leftrightarrow x=W(y)$.

\entry{150412} Deriving Gassner: In \href{http://drorbn.net/AcademicPensieve/2015-04/OneCo.pdf}{2015-04/OneCo.pdf}.

\entry{150409} 2Dv: In \href{http://drorbn.net/AcademicPensieve/2015-04/OneCo.pdf}{2015-04/OneCo.pdf}.

\entry{150107} $\Box\colon\calA(G)\to\calA(G)\otimes\calA(G)$ wrong sketch:
$\bullet$ If $V$ is doubly filtered, the associated graded of the diagonally-associated single filtration of $V$ is isomorphic to the diagonal single-gradation of the associated doubly-graded of $V$. {\bf False.} Take $V=\bbQ\langle x,y\rangle$, $F_{0,0}=F_{1,0}=F_{0,1}=V$, $F_{2,0}=\langle x\rangle$, $F_{1,1}=F_{0,2}=\langle y\rangle$. Then $0+\langle[x]\rangle = V_{1,0}\oplus V_{0,1} \neq V_1 = 0$.
$\bullet$ $\calA(G\times H)\cong\calA(G)\otimes\calA(H)$ as the associated single filtration of the double filtration of $\bbQ(G\times H)$ is its single filtration.
$\bullet$ $g\mapsto (g,g)$ induces $\Box\colon\calA(G)\to\calA(G\times G)\cong\calA(G)\otimes\calA(G)$.

\entry{140723} w-meaning for $\sigma_{ij}\mapsto\small\begin{pmatrix}1-t_j&1\\t_i&0\end{pmatrix}$? u-meaning for $\sigma_{ij}\mapsto\small\begin{pmatrix}1-t_i&1\\t_i&0\end{pmatrix}$? Using the ``other'' Artin rep.\ \bbs{Dalvit}{150318}{160056}?

\entry{150307} Georgetown vocabulary: control theory, zinbieL algebra, Fliess operators, shuffle algebra, dendriform algebra.

\entry{150227} {\bf Infinitesimal $G=\langle X_i\mid R_j\rangle$ definitions}
[\pdftooltip{{\red Br}}{Brochier's MO question}\href{http://mathoverflow.net/questions/69541/almost-direct-product-and-1-formality}{$\to$}],
[\pdftooltip{{\red DPS}}{Dimca, Papadima, Suciu: Topology and Geometry of Cohomology Jump Loci, arXiv:0902.1250}\href{http://arxiv.org/abs/0902.1250}{$\to$}].
$\bullet$ Pro-unipotent?
$\bullet$ Malcev completion: $\Mal(G) \coloneqq \varprojlim\bbQ \!\otimes_\bbZ\! (G/G^{(n)})$.
$\bullet$ $\gr G \coloneqq \bbQ\otimes_\bbZ\bigoplus G^{(n)}/G^{(n+1)}$.
$\bullet$ Malcev Lie algebra: roughly, $\mal(G) \coloneqq \hat{\FL}(x_i) / (\log R_j)$, with $x_i\coloneqq\log X_i$. Is filtered.
$\bullet$ 1-formal: $\mal(G)$ isomorphic as filtered to a quadratic Lie algebra.
$\bullet$ Holonomy Lie algebra of $X$: $\sim$ quadratic generated by $H_1$ modulo $\operatorname{im}H_2$.

\entry{150224a} Surface braids: Bardakov, Bellingeri, Birman, Funar, Gervais, Gonzalez-Meneses, Guaschi, Juan-Pineda.

\entry{141226a} With Dalvit: for $(+,+)\neq(s_1,s_2)\in\{\pm\}^2$, is there $\phi\in \Aut(\FG(x,y))$ s.t.\ $\phi(y^{-1}xy) = y^{-s_1}xy^{s_1}$, $\phi(x^{-1}yx) = x^{-s_2}yx^{s_1}$?
Sela: $\Out(\FG_2)\coloneqq\Aut(\FG_2)/\Inn(\FG_2) = \Aut(\FG_2/[\FG_2,\FG_2])=GL_2(\bbZ)$, hence easily not.
Chterental: That's easily within ``Whitehead's algorithm''.
Why bother? Otherwise the 4 distinct handshake w-links of \bbs{Dalvit}{140617}{110606} could be equal as 2-knots contradicting Satoh's conjecture \& showing that $Z^w$ doesn't extend via BF.

\entry{150219} Jones ribbon conditions from the Oberwolfach-1405 AKT?

\entry{150206} Study annular braids / tangles. Canonical forms?

\entry{131104} Humbert's thesis pp 22: The relations of ${\mathfrak t}^1_n$: $[v_i,w_j]=\langle v,w\rangle t_{ij}$, $[v_i,t_{jk}]=0$, $[x_i,y_i]=-\sum_{j\neq i}t_{ij}$. Imply centrality of $\sum_j v_j$ and $t_{ij}=t_{ji}$, $[v_i+v_j,t_{ij}]=0$, $[t_{ij},t_{kl}]=0$, and $[t_{ij},t_{ik}+t_{jk}]=0$. Canonical forms?

\entry{150217} Enriquez: $B^1_n$ is $\langle\sigma_i,X^\pm_1\rangle$ mod $(\sigma_1^{\pm1}X_1^\pm)^2=(X_1^\pm\sigma_1^{\pm1})^2$, $[X_1^\pm,\sigma_i]=1$ for $i\geq 2$, $[X_1^-,(X_2^+)^{-1}]=\sigma_1^2$, $X_1^\pm\cdots X_n^\pm=1$, and braid relations, where $X_{i+1}^\pm=\sigma_i^{\pm 1}X_i^\pm\sigma_i^{\pm 1}$.

\entry{150210} Reidemeister-Schreier: 1. $H<G$ $\leadsto$ groupoid $H\backslash G$ with objects cosets $H\gamma$, morphisms $(H\gamma,g)\colon H\gamma\to H\gamma g$, and compositions $(H\gamma,g_1)\act(H\gamma g_1,g_2)\coloneqq(H\gamma, g_1g_2)$. With this, $H=\Aut(He)$. 2. If $G=\langle X\colon R\rangle$, $H\backslash G$ is presented with $X\times(H\backslash G)$ generators and $R\times(H\backslash G)$ relations. 3. There's a same-size presentation of $\Aut(He)$.

\entry{150208} Kohno knew elliptic KZ in 1996.

\vskip -2mm
\needspace{5mm} % 4mm is not enough.
\parpic[r]{\includegraphics[height=0.6in]{../../2015-01/thumbs/Order4Torus.png}}
\entry{150201a} In \href{http://drorbn.net/AcademicPensieve/2015-01/}{2015-01}:

\entry{141204a} {\bf Prob.} Find a simple description of simple 2-knots. Done in Kawauchi's {\em\href{http://www.sci.osaka-cu.ac.jp/~kawauchi/AChordRibbonSurfaceLink.pdf}{A Chord Diagram of a Ribbon Surface-Link}?}

\entry{150131} Katz 5.1: $R\calA^s(\mid_h\uparrow^n) \hookrightarrow \calA^s(\mid_h\uparrow^n)/C$. $R$: nothing on last strand. $\mid_h$: a handle line.

\entry{150130b} Katz 5.2: $L\calA^{s\Yup}_1(\uparrow^n)\cong\calA^{s\Yup}_1(\uparrow^n)/C$. $\Box_1$: elliptic. $\Yup$: strutless. $s$: skeleton-connected. $L$: only lonely vertices on last strand. $C$: closed surface.

\entry{150130a} {\bf Q.} Why is $\PB^g$ related to non-tangential differential operators on $\Fun(\frakg^g)$?

\entry{141224} Katz points (\bbs{Katz}{141224}{125936}, \arXiv{1412.7848}): $\bullet$ Cheptea-Habiro-Massuyeau's \arXiv{math/0701277} has a Clifford-like relation in sec.~8 (earlier, in Habiro's \arXiv{math/0001185}, fig.\ 48). $\bullet$ LMO for Lagrangian cobordisms partially interprets leg-gluing in $\calB$. $\bullet$ $\calB^g$-grading: \# of trivalent vertices (excluding univalents).

\entry{150123} $\gr\left(\PB_n^0\to\PB_n^1\right)$ is $0\colon\calA^{pb,0}\to\calA^{pb,1}$ for a degree mismatch. Likely $[\PB_n^1,\PB_n^1]\varsupsetneq\PB_n^0$.

\entry{150121} Quillen: $\calU(\bbQ\otimes\gr G)\cong\gr\bbQ G$, where $\gr G$ uses lower central series, and $\gr\bbQ G$ uses the augmentation ideal.

\entry{150112} Whitney's trick, loosely: In high dimensions at $\pi_1=0$, algebraic intersection numbers have precise geometric realizations.

\entry{141221} A-S super-CS: (uncertainties \yellowt{highlighted}) \hfill ($d=\theta^\mu\partial_\mu$)
\[ \calA(\theta)=c + \theta^\mu A_\mu +
  \theta^\mu\theta^\nu\epsilon_{\mu\nu\rho}\partial^{(0)}_\rho\bar{c} +  \text{\yellowm{\theta^\mu\theta^\nu\theta^\rho\epsilon_{\mu\nu\rho}\phi}},
\]
\[ \text{\it SCS}(\calA) = \int dxd\theta\,\tr\left(
    \frac12\calA\cdot d^{(0)}\calA+\frac16\calA^3
  \right).
\]

\vskip -2mm
\needspace{6mm} % 5mm is not enough.
\parpic[r]{
  \includegraphics[width=0.8in]{figs/Baguenaudier.pdf}
  %\includegraphics[height=0.75in]{figs/ChineseRings.jpg}
}
\entry{150106} Przytycki, Sikora's \arXiv{math/0007134} ``Chinese Rings'', \href{https://en.wikipedia.org/wiki/Baguenaudier}{Wikipedia: Baguenaudier}:

\entry{141226c} Presentations of $[\FG,\FG]$, $[\FL,\FL]$?

\entry{141226d} {\bf Q.} If $G\leadsto \gr G=\bigoplus I^n/I^{n+1}$ is understood, is $\gr_2G\coloneqq\bigoplus I^{2n}/I^{2(n+1)}$ interesting? ({\bf A.} Likely not.).

\entry{141209} {\bf Plan.} Understand simple circle-pair diagrams, then attempt to generalize to ones with intersections. {\em Diagrams:} Planar multiple paired oriented circles, AS in said orientation. {\em Relations:} subdivision, $4T_{1,2}$ as in \bbs{Dalvit}{141212}{160547}. Relation with $\calA^w$ at \bbs{Dalvit}{141217}{145942}.

\entry{141208} {\bf Proj.} Milnor/trees / Alexander/MVA / $\pi_1$ for 2-knots with boundary.

\entry{140210} {\bf Proj.} A quick paper on a quick combinatorial construction of the wheels invariant following \href{\myurl/Talks/Hamilton-1412}{Talks: Hamilton-1412}.

\entry{141204b} {\bf Proj.} Write up ``combinatorial KV''$\Rightarrow$``convolutions''.

\entry{141127a} {\bf Q.} Are intersection graphs mod 4T the gr of something?

\entry{141127b} Repeat talks: Watch previous video, repartition handout.

\entry{141114b} {\bf Proj.} Exposition of Enriquez' solution of YB.

\entry{141113a} Boden: Brandenbursky has 2 Alexander polys on $v\!\calK$.

%mathtools: $\prescript{z}{x}{\text{\huge$\overcrossing$}_y^w}$
%$\sideset{_a^b}{_c^d}{\text{\huge$\overcrossing$}}$
%\vskip -1mm
\entry{140726} Boden's $\text{\it VG}_K$: $[s,q]=1$ and
\hfill $\tensor*[_x^z]{\text{\huge$\overcrossing$}}{_y^w} \raisebox{2mm}{$\to$} \begin{array}[b]{l} z=xysx^{-1}s^{-1} \\ w=sxs^{-1} \end{array}$
\newline\null\hfill $\tensor*[_y^w]{\text{\huge$\undercrossing$}}{_x^z} \raisebox{2mm}{$\to$} \begin{array}[b]{l} z=s^{-1}x^{-1}syx \\ w=s^{-1}xs \end{array}$
\hfill $\tensor*[_x^z]{\text{\huge$\virtualcrossing$}}{_y^w} \raisebox{2mm}{$\to$} \begin{array}[b]{l} z=q^{-1}yq \\ w=qxq^{-1} \end{array}$

At $q=1\neq s$, not basis-conjugating. At $q=s$, OC holds. At $s=1$ this is Manturov's $\vB_n\to\Aut(F(x_1,\ldots,x_n,q))$:
\[
  \sigma_i\mapsto\left\{{
    x_i\mapsto x_ix_{i+1}x_i^{-1} \atop x_{i+1}\mapsto x_i
  }\right.
  \text{ or }\left\{{
    x_i\mapsto x_iq^{-1}x_{i+1}qx_i^{-1} \atop x_{i+1}\mapsto qx_iq^{-1}
  }\right.
\]
\[
  \tau_i\mapsto\left\{{
    x_i\mapsto qx_{i+1}q^{-1} \atop x_{i+1}\mapsto q^{-1}x_iq
  }\right.
  \text{ or }\left\{{
    x_i\mapsto x_{i+1} \atop x_{i+1}\mapsto x_i
  }\right.
\]

\entry{141102a} Zung's visit: $\bullet$ We don't fully understand Configuration Space Integrals (CSI) for curves in the punctured plane. $\bullet$ It's likely that every CSI has a Gauss Diagram Formula (GDF), as winding numbers are computable as intersection numbers. $\bullet$ Degree $n$ GDFs are FT of type $\frac32n$, with simple Weight Systems (WS). Likely they are not determined by their WS. $\bullet$ The Merkov quotient of the Feynman-diagram space $\calA^t$ is: $\ast$ internal trivalent vertices vanish. $\ast$ ``Split'' arrow-exchange relation. $\bullet$ Is there a $\delta$ like in KBHs?

\vskip -3mm
\entry{140923} Manturov's rep.\ for $\PvB_n$:
$\displaystyle
  \sigma_{ij}\mapsto\left\{{
    x_i\mapsto qx_iq^{-1} \atop x_j\mapsto x_i^{-1}q^{-1}x_jqx_i
  }\right.$
(and hence there's a map $\PvB_n\to\PwB_{n+1}$).

\entry{141102b} Assaf's riddle: $k$ kids share a loot of $n$ indivisible candies. The first proposes a split; if not accepted by a strict majority, she leaves and the second proposes, etc. How is the loot split?

\entry{131213a} {\bf Proj.} G-FT invariants of plane curves, \href{http://drorbn.net/AcademicPensieve/2014-01/PlaneCurves.pdf}{2014-01/PlaneCurves.pdf}.

\entry{140821} Fiedler: There may exist a ``new''$^*$ non-oracle map $\calK\to\bbZ\calK$. (*) Poly-time, multi-local, low profile, high rank.
% $K5_2\mapsto 2K3_1-2K0_1-K8_{20}+K7_3$ retracted by email of 140928.

\entry{140909} {\bf Question.} Is 2-component 2D linking in 4D non-trivial, modulo subdivision and melding? {\bf A.} Likely trivial.

\entry{131112b} A map $\calA^w\left({\red\uparrow}\!\multimapdotbothvert\right)\to\calA^u\left(\uparrow\!\oast\right)$ arises in deducing wheeling from the full Duflo (not $\alpha^{-1}$ for $\alpha$ is not well-defined!{\red ?}); There's a pairing $\calA^w\left({\red\uparrow}\!\multimapdotbothvert\right) \otimes \calA^u\left(\uparrow\right) \to \calA^u\left(\uparrow\right)$. Topological meaning?

\entry{140831} {\bf Proj.} Paper: ``Why I care about virtual knot theory?''.

\entry{140731} {\bf Proj.} Make the polynomiality of $B_n$ ridiculously easy.

\entry{140725} Two permutations to the virtual braid: $S_n \xleftarrow{s} \PvB_n \xrightarrow{\varsigma} S_n$ via $(e,\tau_i,(ij))) \xleftarrow{s} (\sigma_i,\tau_i,\sigma_{ij}) \xrightarrow{\varsigma} (\tau_i,\tau_i,e)$.

\entry{140708} Fox $\partial_i\colon F_n\to\bbZ F_n$: $\partial_ix_j=\delta_{ij}$, $\partial_i(uv)=\partial_iu+u\partial_iv$ (a 1-cocycle). Gassner: $b\mapsto\pi\partial_jb(x_i)$, with $\pi\colon\bbZ F_n\to\bbZ\bbZ^n$ the Abelianization.

\entry{140622} Burau: $\Sigma\coloneqq D^2\setminus\{n\text{ pts}\}$, $p\colon\tilde{\Sigma}\to\Sigma$ its $\bbZ$-cover w/ basic deck transformation $t$, $1\in\partial\Sigma$ a basepoint, $\tilde{1}\in\tilde{\Sigma}$ a lift, $1^\ast=p^{-1}(1)$ all lifts, $\tilde{H}\coloneqq H_1(\tilde{\Sigma},1^\ast;\bbZ)$ is a $(\Lambda\coloneqq\bbZ[t^{\pm 1}])$-module. $Bu\colon B_n\to\Aut_\Lambda(\tilde{H})$ is Burau.

\entry{140721} {\bf Proj.} Hilden braids: expansions, the $a/\alpha$-map, tangles?

\entry{140716} The $\bar\mu$ invariants are (homology-) concordance invariant.

\entry{140713} {\bf Proj.} Low v-algebra: Lie bi-algebras \& arrow diagrams.

\entry{140604} In $\bbR^4$, framing a hoop is whatever makes tubing well defined, framing a balloon is whatever makes doubling well defined, and framing a vertex is the interaction between the two.

\entry{140424} Mathematica-WikiLink re-implementation: \newline
{\tt CreateWikiConnection}, {\tt WikiUserName}, {\tt WikiGetPageText}, {\tt WikiSetPageText}, {\tt WikiSetPageTexts}, {\red\tt WikiUploadFile}.

%\vskip -4pt
\needspace{2mm} % 1mm is not enough.
\parpic[r]{\includegraphics[height=12mm]{figs/q-alg-9709040-KontsevichPropagator.png}}
\entry{140422a} The Kontsevich propagator $d\phi$.

\entry{140422b} Find $\int_{\bbC_z} \bigwedge_{i=1}^n d\Arg(z-z_i)\in\Omega^{n-2}(\bbC^n_{z_i})$.

\entry{140422c} Khovanskii's ``On a Lemma of Kontsevich'' proves Kontsevich's vanishing lemma in 3 pages.

\entry{140309} {\bf Proj.} Low degree BF.

\entry{140419} {\bf Proj.} Too many definitions of the Alexander polynomial.

\entry{140417a} Kervaire: $G$ is an ($n\geq 3$)-knot group iff it is f.p., normally generated by one element, and $H_{1,2}(G)=(\bbZ,0)$.

\entry{140417b} Does every simple decker set come from a (ribbon) 2-link? Is a 2-link group a LOF group? A decker group a 2-link group?

\entry{140413} Are there ``spherical w-braids''?

\entry{140316} {\bf Proj.} What are all internal quotients of $\FL$ (compare ``PI-Rings'')? Which are of polynomial growth?

\entry{140325} Monty Hall: A prize is in 1 of 3 envelopes. You choose one, an oracle shows another to be empty. Will you switch? Deliberate oracle: Yes. Chance oracle: No. What makes it so confusing?

\entry{140318} {\bf Proj.} Study Vogel's weight system in the context of $\calA^v$.

\entry{140304} Itai's iterated mean value theorem: with $(\delta f)(x)\coloneqq f(x+1)-f(x)$, $\forall x_0,n\,\exists x\in[x_0,x_0+n]\,\text{s.t.}\, (\delta^nf)(x_0)=(\partial^nf)(x)$.
{\em Pf.} With $\chi\coloneqq 1_{[0,1]}$, $\delta=\chi\star\partial$ hence $\delta^n=\chi^{\star n}\star\partial^n$ and as $\|\chi^{\star n}\|_{L^1}=1$, $(\delta^nf)(x_0)$ is bound between the extremals of $(\partial^n f)(x)$.

\entry{140302} Assaf: The fundamental group and the fundamental groupoid of a path-connected space are naturally equivalent.

\entry{140228b} What's the relation between quandle cocycles and 2-knots?

\entry{140227} Itai: For at least some quadruples of lines in $\bbR^3$, there are at least two lines that intersect all of them.

\entry{140106} Is the $\vee$-invariant of gnots trivial on 2-knots? Is there a multiplicative Alexander duality? Alexander: $X\subset S^n$ compact, locally contractible $\Rightarrow$ $H^q(X)\simeq H_{n-1-q}(X^c)$.

\entry{140218a} Vienna vocabulary: cobordism hypothesis, WKB approximation, Fukaya category, gerbes, fusion categories, differential cohomology.

\entry{140218b} {\bf Proj.} Clean and write up the shielding story.

\entry{140218c} {\bf Proj.} A note on how $DG$ arises in the context of KBHs.

\entry{140217} \href{http://ncatlab.org/nlab/show/Drinfeld+center}{nLab}: $\calC$ monoidal category. Its Drinfel'd centre is the BMC with objects pairs $(X,\beta)$ of $X\in\calC$ and natural isomorphism $\beta_-\colon X\otimes(-)\to(-)\otimes X$ such that $\forall Y,Z\in\calC,\, \beta_{Y\otimes Z}=(I_Y\otimes\beta_Z)\act(\beta_Y\otimes I_Z)$, with $\Hom((X,\beta),(X',\beta'))\coloneqq\{f\in\Hom(X,X')\colon\forall Z,\,\beta_Z\act(I_Z\otimes f)=(f\otimes I_Z)\act\beta'_Z\}$, $(X,\beta)\otimes(X',\beta')\coloneqq(X\otimes X',(I_X\otimes\beta')\act(\beta\otimes I_{X'}))$, and $R_{(X,\beta),(X',\beta')}\coloneqq\beta_{X'}$ (?).

\entry{140211} Ogasa's {\em Local Move Identities\ldots} --- some skein relations for high-dimensional Alexander.

\entry{140120} How exactly do normal Euler numbers relate to branch points? Can the latter be avoided?

\vskip -2mm\needspace{0mm}
\parpic[r]{\includegraphics[height=8.5mm]{figs/SatohExample5-10.png}}
\entry{140130} Satoh's w-knot has the same $\pi_1$ and the same $Z$-polynomial (Sawollek) as the trefoil.

\entry{140115} Chterental: $\vB_n$ acts faithfully on ``virtual curve diagrams'', and with run-length compression, this is describable in poly time.

\entry{140126} Is tube-bypass an unknotting operation for 2-knots?

\entry{140117} Carter: A spun Hopf link with an additional orthogonal plane running once above and once below it makes a knotted $2T^2+S^2$ with 4 triple points.

\entry{140116} Using LMO, FT invariants of links in $S^3$ extend to links in arbitrary $\bbQ HS$. A simple description? \refentry{190321}

\entry{140114} Gavish: ``Singular value decompositions''.

\entry{140113} Many papers by Seiichi Kamada.

\entry{131229} Smale (`57): Long immersions $\bbR^k\hookrightarrow\bbR^m$ are classified by $\pi_k(V_{m,k})$ where $V_{m,k}$ is the Stiefel manifold (linear embeddings $\bbR^k\hookrightarrow\bbR^m$). Paechter (I, `56): $\pi_2(V_{4,2})=\bbZ$.

\entry{131219} $Gr(\bbR^2\hookrightarrow\bbR^4)=S^2\times S^2$. Yael:
$\bullet$ There's a $\bbC P^2=S^2$ of complex lines in $\bbC\times\bbC$ and in $\bbC\times\bar\bbC$.
$\bullet$ It is the product of the moduli $C(\bbR^4)\times C(\bar\bbR^4)$ of metric complex structures on $\bbR^4$ / $\bar\bbR^4$. For $(I,\bar I)\in C(\bbR^4)\times C(\bar\bbR^4)$ there is a unique $\bbR^2\hookrightarrow\bbR^4$ which is complex relative to both, and a given $P=\bbC=\bbR^2\hookrightarrow\bbR^4$ determines two metric complex structures on $\bbR^4/\bar\bbR^4$ by multiplication by $i$ on $P$ and by $\pm i$ on $P^\perp$. Finally $C(\bbR^4)=\text{\it SO}(4)/U(2)=\{\text{left multiplications $L_u$ by unit imaginary quaternions $u$}\}=S^2$ and $C(\bar\bbR^4)=\{R_v\}_{v\in S^2\subset\bbR^3\subset\bbH}$.
$\bullet$ $P(u,v)=\Span(u+v,uv-1)$ or $\Span(u-v,uv+1)^\perp$ and for orthonormal $(\alpha,\beta)$, $\Span(\alpha,\beta)\mapsto(\beta\bar\alpha,\bar\alpha\beta) = ((\alpha\wedge\beta)^+,(\alpha\wedge\beta)^-)$, the last using the self-dual and anti-self-dual projections $\Lambda^2\to\Lambda^{2\pm}$.

\entry{131218} Bjorndahl: $N$ prisoners each wears $\infty$-many b/w hats. Simultaneously each needs to point at a black hat on her head. How can they maximize the chance that they will {\em all} get it right?

\entry{131217} Goryunov's {\em finite order \ldots $J^+$ \ldots}: Generic smooth plane curves, allowing triple points and opposite self-intersections, map to knots in the solid torus $ST^\star\bbR^2$ inducing an isomorphism of projectivizations.

\entry{131213b} Stallings' theorem: $h\colon A\to B$ a group homomorphism w/ $h\colon H_1(A)\simeq H_1(B)$ and $h\colon H_2(A)\twoheadrightarrow H_2(B)$. Then $h\colon A/A_n\simeq B/B_n$, where $A_n$, $B_n$ denote lower central series (+ more\ldots).

\entry{131213c} Getzler (bbs): Homotopy 2-types are determined by the action $\pi_1\lefttorightarrow\pi_2$ and a class in $H^3(\pi_1,\pi_2)$.

\entry{131126b} Anton: is there a triality for solutions of the KV equation? --- Yes, {\tt 2013-11/DoubleTree/TrialityComputations.nb}. Minor: does $\alpha$ intertwine the triality of $\sder_2$ with that of $\tder_2\ltimes\attr_2$? --- {\tt 2013-12/}: Most likely not.

\entry{131211} Is the nilpotent completion of the fundamental group of a gnot complement always free-nilpotent? --- No; Abelian it is for $\bbR^2\times\{0\}\cup\{0\}\times\bbR^2$.
%$\{(x,y,0,0)\}\cup\{(0,0,z,w)\}$.

\entry{131122a} {\bf Proj.} Figure out the bubble-wrap-finite-type invariants of {\em all} knotted objects in $\bbR^4$.

\entry{131202a} From {\em Virtual 2-Knots} by Schneider:

\hfill\includegraphics[height=1in]{figs/Virtual2KnotsBySchneider-1.png}
\hfill\includegraphics[height=1in]{figs/Virtual2KnotsBySchneider-2.png}
\hfill\null

In light of ``virtual doodles'', perhaps this should be modified?

\entry{131205a} Are there 3 embedded surfaces in $\bbR^4$ so that any 3 immersed handlebodies bounding them have a common point?

\entry{131205b} Cimasoni: Levine's {\em Poly.\ Inv.\ of Knots of Codimension 2.}

\entry{131204} Coboundary: $\delta(x)=[r,\Delta(x)]$, with invariant $r+r^{21}$.

\entry{131202b} Farber`s {\em Noncommutative Rational Functions and Boundary Links,} continued Retakh, Reutenauer, Vaintrob, \arXiv{math/0004112}.

\entry{131130c} Yanagawa (`69): Ribbon 2-knot $K$ is trivial iff $\pi_1(K)\simeq\bbZ$.

\entry{131130d} Meilhan: 2-knots papers by Yajima (`62, `64), Yanagawa (`$69^3$), Omae (`71).

\entry{131126a} Crainic \arXiv{math/0403266} on Homological perturbations: A {\em Homological Homotopy Equivalence (HHE)} is a pair of complexes with quasi-isomorphisms $\xymatrix{(L,b) \ar@<2pt>[r]^i & (M,b) \ar@<2pt>[l]^p}$, with a homotopy $h$ between $1=1_M$ and $ip$, so $ip=1+bh+hb$. A {\em perturbation} is $\delta\colon M\to M$ with $\deg b=\deg\delta$ and $(b+\delta)^2=0$; it is {\em small} if $(1-\delta h)^{-1}$ exists. {\bf Claim.} $\xymatrix{(L,b_1) \ar@<2pt>[r]^{i_1} & (M,b+\delta) \ar@<2pt>[l]^{p_1}}$ is again an HHE, with $A\coloneqq(1-\delta h)^{-1}\delta$, $b_1\coloneqq b+pAi$, $i_1\coloneqq i+hAi$, $p_1\coloneqq p+pAh$, and with $h_1\coloneqq h+hAh$.

\entry{131212} Cimasoni: There is a 1-double-point gnot.

\entry{131209} Cimasoni: There is a natural smoothing of 2-gnots.

\vskip -2mm\needspace{11mm} % 10mm is not enough.
\parpic[r]{$\xymatrix@R=7mm{
  D^2 \ar[dr]^(.3)\beta & S\!O(3) \ar[d]^{e_{1}} \\
  S^{1} \ar@{^{(}->}[u] \ar[r]^\gamma \ar[ur]^(.3)\phi & S^{2}
}$}\picskip{4}
\entry{131122b} Moskovich, \arXiv{math/0211223}: On the right, $\phi$ and $\beta$ pair to an integer. Indeed $D^2\ni x\mapsto\beta(x)^\perp$ is a circle bundle on $D^2$ which must be trivial, inducing a trivialization of the circle bundle $S^1\ni x\mapsto\beta(x)^\perp$. But $\phi\act e_2$ is a section of that bundle, hence an integer.

\entry{131121} Burke, Koytcheff: \arXiv{1311.4217}, {\em A colored operad for string link infection}.

\entry{131114} Bar-Hillel's Simpson's paradox: In Israel in every age bracket death rates for Arabs are higher than for Jews, yet overall death rates for Jews are higher.

\entry{131111} Massuyeau (bbs). Given an algebra $A$ and $N\geq 1$, $\exists$ commutative algebra $A_N$ s.t.\ $\forall$ commutative algebra $B$,
\[ \Hom_\text{Alg}(A,\Mat_N(B))\simeq\Hom_\text{C-Alg}(A_N, B). \]
\vskip -3mm
Indeed
\vskip -6mm
\[ A_N\simeq\frac
  {\langle a_{ij}\colon a\in A,\ 1\leq i,j\leq N\rangle}
  {(a+\lambda b)_{ij}=a_{ij}+\lambda b_{ij},\ 1_{ij}=\delta_{ij},\ (ab)_{ik}=\sum_ja_{ij}b_{jk}}.
\]

\entry{131110} Massuyeau (bbs, eprints), after Van den Bergh: A double bracket in an algebra $A$ is $\llbracket-,-\rrbracket\colon A\otimes A\to A\otimes A$ s.t.
(1) $\llbracket b,a\rrbracket=-\llbracket a,b\rrbracket^{op}$.
(2) $\llbracket a,b_1b_2\rrbracket=(b_1\otimes 1)\llbracket a,b_2\rrbracket + \llbracket a,b_1\rrbracket(1\otimes b_2)$.
It is Poisson if

\vspace{-4mm}
\[ \llbracket-,-,-\rrbracket \coloneqq
  \begin{array}{c}\includegraphics[width=1.5in]{figs/DoubleBracketJacobi.pdf}\end{array}
  =0
\]
\vskip -1mm

\entry{131107} Enriquez/EllipticAssociators: with $\bar{e}(z)\coloneqq\frac{\ad z}{e^{\ad z}-1}$,
\begin{align*} (\mu,\Phi)\mapsto
  A&=\Phi\left(-\bar{e}(x)y,t^{12}\right)
    e^{-\mu \bar{e}(x)y}\Phi^{-1}\left(-\bar{e}(x)y,t^{12}\right), \\
  B&=e^{\mu t^{12}/2}\Phi\left(\bar{e}(-x)y,t^{21}\right)
    e^x\Phi^{-1}\left(-\bar{e}(x)y,t^{12}\right).
\end{align*}

\entry{131109} C.\ Frohman, A.\ Nicas, {\em The Alexander Polynomial via topological quantum field theory,} Differential Geometry, Global Analysis, and Topology, Canadian Math.\ Soc.\ Conf.\ Proc.\ {\bf 12}, Amer.\ Math.\ Soc.\ Providence, RI, (1992) 27--40.

\entry{131106b} Massuyeau (bbs, eprints, easy): $\exists$ ``symplectic expansion'' --- a group-like expansion $Z\colon \FG(x_i,y_i)\to \FA(\bar x_i,\bar y_i)$ with
$ Z\left(\prod_i[x_i,y_i]\right)
  = \exp\left(-\sum_i[\bar x_i,\bar y_i]\right)
$.
Thus surface groups are quadratic and have homomorphic expansions.

\entry{131027a} Cattaneo: ``BV is the `right' de-Rham differential on super-manifolds.''

\entry{131027b} The Hilbert basis theorem: An ideal in the ring of multivariable polynomials over a Noetherian ring is finitely generated.

{\em Pf.} Enough, $R$ Noetherian $\Rightarrow$ any $I\subset R[x]$ is finitely generated. Let $p_n\in I\setminus\langle p_1,\ldots,p_{n-1}\rangle$ be of minimal degree. As $R$ is Noetherian, for large $N$ the leading coefficient of $p_N$ is a combination of previous leading coefficients, so it can be killed off contradicting the minimality of $P_N$. $\Box$\quad Can be made constructive using Gr\"obner bases.

\entry{131020} Artin-Wedderburn: A semi-simple ring is uniquely (up to a permutation) isomorphic to a product of finitely many finite matrix rings over division rings.

\entry{131007c} I don't understand \v{S}evera's\hfill
$\xymatrix@C=2.5mm@R=5mm{
  (A\!\otimes\!A)\!\otimes\!A \ar[rr]^\Phi \ar[d]^{m\!\otimes\!1}
  & & A\!\otimes\!(A\!\otimes\!A) \ar[d]_{1\!\otimes\!m} \\
  A\!\otimes\!A \ar[r]^m
  & A
  & A\!\otimes\!A \ar[l]_m
}$

\needspace{2mm} % 1mm is not enough.
\entry{131007b} From 2013-10/Swinging:
\vskip -4mm
\includegraphics[width=0.49\textwidth]{figs/swinging.pdf}

\entry{131007a} Monoblog starts.

\write\posfile{Null]}

\end{multicols*}

\newpage\begin{multicols}{2}

\textbf{Archived Items.} %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%% %%%%%

\archived{231105b} Is ``almost classical'' an appropriate tag for $w$ objects with $\atdiv=0$ (KV solutions, horizontal chord associators)? \refentry{231105a} (After archiving: No.)

\archived{210511} {\bf Chal.} Reconstruct a category like $\{\Hom(V^{\otimes A}\to V^{\otimes B})\}$ from a ``forward contraction algebra'' akin to $\{(V^\ast)^{\otimes A}\otimes V^{\otimes B}\}$ for (some) $\infty D$ vector spaces $V$. {\bf DoPeGDO} Sol'n: L/G variables are chronologically ordered. In $Q$, LL terms are small, and also LG terms if L preceeds G. Allow only these constrained LG contractions.

\archived{210421} {\bf T/F?} $\sqrt{G}^{-1}\Gamma\sqrt{G} = \Gamma^T$.

\archived{170913a} A pushforwards challenge in \href{http://drorbn.net/AcademicPensieve/2017-09/Pushforwards.pdf}{Pushforwards.pdf}.

\archived{230612a} 20m talk: ``Rooting the BKT for FTI''. Key: $X\subset\underbar{n}^d$, $|X|=P$. Using dyadic decompositions, in time $\sim P$ can set a database of size $\sim P$ so $|X\cap R|$ can be computed in time $\sim 1$ for every rectangle $R\subset\underbar{n}^d$.

\archived{230109} {\bf Def.} Given a v.s.\ $V$, a Partial Quadratic (PQ) $Q$ on $V$ is a symmetric bilinear form $Q$ on a subspace $\calD(Q)\subset V$. For $U\subset\calD(Q)$, denote $\ann_Q(U)\coloneqq\{v\in\calD(Q): Q(U,v)=0\}$.

{\bf Def.} $Q_1+Q_2$ is with $\calD(Q_1+Q_2)=\calD(Q_1)\cap\calD(Q_2)$.

{\bf Def.} Given a linear $\psi\colon V\to W$ and a PQ $Q$ on $W$, the pullback is $(\psi^*Q)(v_1,v_2)=Q(\psi v_1,\psi v_2)$ with $\calD(\psi^*Q)=\phi^{-1}(\calD(Q))$.

{\bf Def.} Given $\phi\colon V\to W$ and a PQ $Q$ on $V$ the pushforward $\phi_*Q$ is with $\calD(\phi_*Q)=\phi(\ann_Q(\calD(Q)\cap\ker\phi))$ and $(\phi_*Q)(w_1,w_2)=Q(v_1,v_2)$, where $v_i$ are s.t.\ $\phi(v_i)=w_i$ and $Q(v_i,\rad Q|_{\ker\phi})=0$.

\parpic[r]{$\xymatrix@C=3mm@R=3mm{
  \bullet \ar[r]^\alpha \ar[d]_\gamma & \bullet \ar[d]^\beta \\
  \bullet \ar[r]_\delta \ar@{.>}[ru] & \bullet
}$}
{\bf Thm(?).} $\psi^*$ and $\phi_*$ are well-defined and functorial, and if $\alpha\act\beta = \gamma\act\delta$, then $\gamma^*\act\alpha_* = \delta_*\act\beta^*$. $\psi^*$ is additive but $\phi_*$ isn't. $\phi^*\act\phi_*$ is restriction to $\im\phi$. $\phi_*\act\phi^*$ is ?.

{\bf Thm(?).} Over $\bbR$, given $\phi\colon V\to W$ and PQs $Q$ on $V$ and $C$ on $W$,
\[ \sign_V(Q+\phi^*C) = \sign_{\ker\phi}(\iota^*Q) + \sign_W(C+\phi_*Q) \]
\[ (\Leftrightarrow\ \sign_V(Q) = \sign_{\ker\phi}(\iota^*Q) + \sign_W(\phi_*Q) \quad\text{(no $C$, no $+$)}). \]

\archived{230109} {\bf Def.} Given a v.s.\ $V$, a Partial Quadratic (PQ) on $V$ is $Q=(D=\calD(Q)\subset W,L=\calL(Q)\colon D\to D^*)$ with $L=L^\star$. Write $Q(v_1,v_2)=L(v_1)(v_2)$.

{\bf Def.} Given a linear $\psi\colon V\to W$ and a PQ $Q=(D=\calD(Q)\subset W,L=\calL(Q)\colon D\to D^*,\ L=L^\star)$ on $W$, the pullback $\psi^*Q$ is $(D',L')$ with $D'=\phi^{-1}(D)$ and $L'=\psi\act L\act\psi^*$, so $(\psi^*Q)(v_1,v_2)=Q(\psi v_1,\psi v_2)$.

{\bf Def.} If $Q_i$ are PQs on $V$, $Q_1+Q_2$ is with $\calD(Q_1+Q_2)=\calD(Q_1)\cap\calD(Q_2)$.

{\bf Def.} Given a linear $\phi\colon V\to W$ and a PQ $Q=(D,L)$ on $V$ the pushforward $\phi_*Q$ is $Q'=(D',L')$ with $D'=\phi((L(\rad Q|_{D\cap\ker\phi}))^\bot)$ and $Q'(w_1,w_2)=Q(v_1,v_2)$, where $v_i$ are s.t.\ $\phi(v_i)=w_i$ and $Q(v_i,\rad Q|_{\ker\phi})=0$.

{\bf Thm.} Pullbacks / pushforwards are well-defined and functorial.

{\bf Thm(?).} Over $\bbR$, given $\phi\colon V\to W$ and PQs $Q$ on $V$ and $C$ on $W$,
\[ \sign_V(Q+\phi^*C) = \sign_{\ker\phi}(\iota^*Q) + \sign_W(C+\phi_*Q). \]

\archived{220923} {\bf Q.} For balanced Heisenberg exponentials, does $p$-scattering determine $x$-scattering?

\archived{210101} {\bf Proj.} Implement a {\tt PD2ThinMorse} with guaranteed bounds and/or using simulated annealing.

\archived{211210} {\bf Proj.} Complexity by ropelength?

\archived{180820} $\rho_1 = \left.t\left(P|_{e,l,f\to 0}-t\omega'\omega^3\right)\right/(t-1)^2\omega^2$ and
\[ P = A^2\frac{(t-1)^3\rho_1+t^2(2vw+(1-t)(1-2c))AA'}{(1-t)t}. \]

\archived{210302} {\bf Q.} (w/ Abbasi) Does every contractible but not manifestly contractible curve in the annulus have an odd self-intersection?

\archived{170829a} {\bf Do.} Find Duflo in Goldman-Turaev.

\archived{190108} {\bf Do.} With $P=\sum a_{mn}z^m\zeta^n$, compute $\llangle\epsilon P\rrangle \coloneqq \log\langle\exp\epsilon P\rangle$.

\archived{150416} Chterental: Is there a Melvin-Morton statement for v-knots?

\archived{181031} {\bf Proj.} Verify Kashaev's conjecture @\arXiv{1801.04632}, re.\ Tristram--Levine signatures.

\archived{170321} For NOE1 with $\Lambda\to0$, are there interesting $R$'s?

\archived{170320b} {\bf Proj.} $k$-co inductive constructions.

\archived{210318a} Riba Garcia's \href{https://lrobert.perso.math.cnrs.fr/kos.html}{talk}, ``Invariants of Rational Homology 3-Spheres and the Mod $p$ Torelli Group'':

\archived{210211a-old} Halacheva ($\sim$):
  $\calA(X)\coloneqq\bigoplus_k\End(\Lambda^kX)$
is a traced meta-monoid with
  $m^{xy}_z(A)\coloneqq(z\to y)\act(e_x\act i_x\act A\act e_y\act i_y - e_x\act A\act i_y)\act{(x\to z)}$
and
  $\tr_x(A)\coloneqq e_x\act i_x\act A\act e_x\act i_x - e_x\act A\act i_x$.
Contains $\Gamma$ (w/ fixed colours) via
  $\Upsilon\colon(\omega,M)\mapsto\omega\Lambda^\ast(M)$.
Predict $\calA$ from $\Gamma$? Interpret $\calA$ in $ybax$? Related to super-algebras? Raise $\calA$ to meta-Hopf? Understand $\im(\Upsilon)$?

\archived{180311} {\bf Do.} With strongly docile $L$ and $\Lambda$, compute $\log\langle\bbe^L\mid\bbe^\Lambda\rangle$ without exponentiating.

\archived{171029} {\bf Do.} Solve $\hbar^{-1}(1-\bbe^{\hbar(t-2a\epsilon)}) = g(a-1,z) + (-\bbe^{\epsilon\hbar}-(t-2a\epsilon)\partial_z+\epsilon z\partial_z^2)g(a,z)$.

\archived{170309} \refentry{170309}
Then \bbs{AKT17}{170317}{120957}: $\bbe^{\alpha w}\bbe^{\beta u} = \bbe^{au}\bbe^{d(b-2\epsilon c)}\bbe^{bw}$ with $\gamma=1-\alpha\beta\epsilon$, $a=\beta/\gamma=\beta+\ldots$, $b=\alpha/\gamma=\alpha+\ldots$, $d=\epsilon^{-1}\log\gamma=-\alpha\beta+\ldots$, so
$\displaystyle \bbO\left(wu\colon \bbe^{\alpha w+\beta u}\right)
  = \bbO\left(ucw\colon \bbe^{au+bw+d(b-2\epsilon c)}\right) =$
$\displaystyle \bbO\left(ucw\colon \bbe^{\lambda_\epsilon(\alpha,\beta)}\bbe^{\alpha w+\beta u-\alpha\beta b}\right)
  = \bbO\left(ucw\colon \bbe^{\lambda_\epsilon(\partial_w,\partial_u)}\bbe^{\alpha w+\beta u-\alpha\beta b}\right)$
so $\displaystyle \bbO\left(wu\colon \bbe^{\alpha w+\beta u+\delta uw}\right)
= \bbO\left(ucw\colon \bbe^{\delta\partial_\alpha\partial_\beta} \bbe^{\lambda_\epsilon(\partial_w,\partial_u)} \bbe^{\alpha w+\beta u-\alpha\beta b}\right) =$
$\displaystyle \bbO\left(ucw\colon\bbe^{\lambda_\epsilon(\partial_w,\partial_u)} \nu\bbe^q\right) = \bbO\left(ucw\colon\bbe^{\Lambda_\epsilon}\nu\bbe^q\right)$,
with $\nu=(1+b\delta)^{-1}$, $q=\nu(\alpha w+\beta u+\delta uw-\alpha\beta b)$ and $\Lambda_\epsilon\in\bbQ(w,u,b,c,\alpha,\beta,\delta)\llbracket\epsilon\rrbracket$.

\archived{170522} What is the ``sensical'' sub-meta-object of
\newline\null\hfill$(\calU(\frakb_+), m, \Delta, S, P, R)$?

\archived{180528a} Is there an operation-uniformizing ``bottom tangles in handlebodies'' theory for (rotational) virtuals similar to Habiro-Massuyeau?

\archived{131112a} The diamond lemma: If $\rightarrow$ is a connected Noetherian binary relation (Noetherian: an infinite $a_1\rightarrow a_2\rightarrow\cdots$ is ultimately constant), and if whenever $a\rightarrow b$ and $a\rightarrow c$ there is $d$ with $b\Rightarrow d$ and $c\Rightarrow d$ where $\Rightarrow$ is the reflexive transitive closure of $\rightarrow$, then $\exists! m\,\forall a\, a\Rightarrow m$.

\archived{200204b} {\bf Talk.} Over then Under Tangles.
{\bf Abstract.} Brilliant wrong ideas should not be buried and forgotten. Instead, they should be mined for the gold that lies underneath the layer of wrong. In my talk I will explain how ``over then under tangles'' lead to an easy classification of knots, and under the surface, also to some valid mathematics: \ldots

\vskip -1.5mm\needspace{13mm}
\parpic[r]{\scalebox{0.8}{\input{figs/GLnUL.pdf_t}}}
\archived{170126c} In $gl_{n+}^\epsilon$: $[\uppertriang,\uppertriang]=\uppertriang$, $[\lowertriang,\lowertriang]=\epsilon \lowertriang$, $[\lowertriang,\uppertriang]=\lowertriang+\epsilon\uppertriang$, so with $h_i=h'_i-\epsilon g_i$,
$[h_i,\cdot]=0$,
$[g_i,g_j]=0$,
$[e_{ij},e_{kl}]=\delta_{jk}e_{il}-\delta_{il}e_{kj}$,
$[f_{ij},f_{kl}]=\epsilon(\delta_{jk}f_{il}-\delta_{il}f_{kj})$,
$[e_{ij},f_{kl}] = \delta_{jk}(\epsilon\delta_{i<l}e_{il}+\delta_{i>l}f_{il}) - \delta_{li}(\epsilon\delta_{k<l}e_{kj}+\delta_{k>j}f_{kj}) + \delta_{jk}\delta_{li}((h_i-h_j)/2+\epsilon(g_i-g_j))$,
$[g_i,e_{jk}]=(\delta_{ij}-\delta_{ki})e_{jk}$,
$[g_i,f_{jk}]=(\delta_{ij}-\delta_{ki})f_{jk}$,
$\deg(\epsilon,h_i,f_{ij},g_i,e_{ij})=(1,1,1,0,0)$.
Verification in \href{http://drorbn.net/AcademicPensieve/2017-02/nb/glne.pdf}{2017-02/glne.nb}. Order $n$ symmetry in \href{http://drorbn.net/AcademicPensieve/2020-01/nb/glne.pdf}{2020-01/glne.nb}.

\archived{171012} \href{http://www.math.toronto.edu/~drorbn/Talks/LesDiablerets-1708/}{Talks/LesDiablerets-1708}, esp.\ \href{http://drorbn.net/AcademicPensieve/Talks/LesDiablerets-1708/nb/PBWDemo.pdf}{PBWDemo.nb}, verifications \href{http://drorbn.net/AcademicPensieve/2017-10/nb/Phi2CR-Classical.pdf}{2017-10/Phi2CR-Classical.nb}: In $\hat\calU(\frakg^\epsilon) = \langle t,y,a,x\rangle /
([t,\cdot]=0,\,[a,x]=x,\,[a,y]=-y,\, [x,y]=t-2\epsilon a)$, we have $\prod_{i=1}^2 \bbe^{\tau_i t}\bbe^{\eta_iy}\bbe^{\alpha_ia}\bbe^{\xi_i x} = \bbe^{\tau t}\bbe^{\eta y}\bbe^{\alpha a}\bbe^{\xi x}$, with
\begin{align*}
  \tau = & \tau_1+\tau_2-\frac{\log(1-\epsilon\eta_2\xi_1)}{\epsilon}
    \gray = \tau_1+\tau_2+\eta_2\xi_1 + \frac{\epsilon}{2} \eta_2^2\xi_1^2 + \ldots, \\
  \eta = & \eta_1 + \frac{\bbe^{-\alpha_1}\eta_2}{(1-\epsilon\eta_2\xi_1)}
    \gray = \eta_1+\bbe^{-\alpha_1}\eta_2 + \epsilon\bbe^{-\alpha_1}\eta_2^2\xi_1 +\ldots, \\
  \alpha = & \alpha_1+\alpha_2 + 2\log(1-\epsilon\eta_2\xi_1)
    \gray = \alpha_1+\alpha_2 -2\epsilon\eta_2\xi_1 + \ldots, \\
  \xi = & \frac{\bbe^{-\alpha_2}\xi_1}{(1-\epsilon\eta_2\xi_1)} + \xi_2
    \gray = \bbe^{-\alpha_2}\xi_1+\xi_2 + \epsilon\bbe^{-\alpha_2}\eta_2\xi_1^2 + \ldots.
\end{align*}

\archived{181024} With Ens. The $C\!D_a$ universe $\calU = \FA\langle H_k,R_k,\dots,Z^i_k,\dots\rangle\ast\bbQ S_{\!\ast}$ has a strand-filtration $\calF_n$, a Vassiliev degree $\deg\geq 0$, a homological degree $\hgt\geq 0$, an $\hgt$-odd differential $\delta$ with $\deg\delta=0$, $\hgt\delta=-1$, an endomorphism $c$ with $c\calF_n\subset\calF_{n+1}$ and $\deg c=\hgt c=0$ and
\begin{itemize}[leftmargin=*,labelindent=0pt,itemsep=-4pt,topsep=0pt]
\item $S_{\!\ast}\coloneqq\bigcup_{n>0}S_{\!n}$ with $(\deg,\hgt)=(0,0)$, $S_{\!n}\subset\calF_n$, and $c\colon S_{\!n}\to S_{\!n+1}$ via $(c\sigma)_1=1$ and $(c\sigma)_{i>1}=\sigma_{i-1}+1$.
\item For $U$ any $H$, $R$, or $Z$, $U_k=c^kU_0\eqqcolon c^kU$.
\item $H\in\calF_1$ with $(\deg,\hgt)=(1,0)$. Let $t_{0i}\coloneqq(1i)H(1i)$, let $t_{12}\coloneqq H_1-(12)H(12)$ and at $j>i>0$ let $t_{ij}=(1i)(2j)t_{12}(2j)(1i)$.
\item $R\in\calF_2$ with $(\deg,\hgt)=(1,1)$, $\delta R=\dots$.
\item \ldots
\end{itemize}
Claim/goal: $H_0(\calU,\delta)\cong S_{\!\ast}\ltimes DK_{\{0\}\sqcup\ast}$
%via $H_k\mapsto\sum_{i=1}^{k+1}t_{0i}$,
and $H_1(\calU,\delta)=0$.

\archived{190115} The monoidal category with objects $\bbN$ generated by $\sigma\in\Aut(2)$ with $1_1\otimes\sigma\otimes 1_1 = \sigma\otimes 1_1\otimes\sigma$, possibly with $\sigma^2=1_2$.

\archived{171104} Roland: Solve $g(a,t)g(-a-1,-t)=P(a,t)$.

\archived{170725} {\red (Wrong, see \href{http://drorbn.net/AcademicPensieve/2017-10/nb/Phi2CR.pdf}{2017-10/Phi2CR.nb})} \href{http://drorbn.net/AcademicPensieve/2017-07/nb/Multi-beta-yax.pdf}{2017-07/Multi-beta-yax.nb}: In $\calU_{\gamma^{-1};\gamma\beta}$ where $q=\bbe^\beta$, $\prod_{i=1}^2 \bbe^{\eta_iy}\bbe^{\alpha_ia}\bbe^{\xi_i x} = \bbe^{\eta y}\bbe^{\alpha a}\bbe^{\xi x}\bbe^{\tau t}$, with
\begin{eqnarray*}
\eta & = &
  \eta_1+\eta_2 e^{-\gamma\alpha_1}-\beta\gamma\eta_2^2\xi_1 e^{-\gamma\alpha_1} + \ldots
  = \eta _1 + \delta\eta _2 e^{\beta -\alpha _1 \gamma } \\
\alpha & = &
  \alpha _1+\alpha _2+2 \beta  \eta _2 \xi _1 + \ldots
  = \alpha _1+\alpha_2 - 2 \left(\beta+\log \delta \right)/\gamma \\
\xi & = &
  \xi _1 e^{-\gamma\alpha _2} + \xi_2 - \beta  \gamma  \eta _2 \xi_1^2 e^{-\gamma\alpha _2} + \ldots
  = \delta\xi _1 e^{\beta -\alpha _2 \gamma }+\xi _2 \\
\tau & = &
  -\eta _2 \xi_1 + \beta\eta_2 \xi _1 \left(\gamma  \eta _2 \xi _1+1\right)/2 + \ldots
  = \left(\beta+\log \delta \right)/(\beta\gamma)
\end{eqnarray*}
and $\delta \coloneqq \left(\left(e^{\beta }-1\right) \gamma  \eta_2 \xi _1+e^{\beta }\right)^{-1} = 1-(1+\gamma\eta_1\xi_1)\beta+\ldots$.

\archived{170805} With $\Phi=(\phi_j(\alpha_i))$ and $Z=\zeta(\partial_{\alpha_i})$, set $\Phi_\ast Z \coloneqq \left. \bbe^{\sum\partial_{\beta_j}\phi_j(\partial_{a_i})}\zeta(a_i)\right|_{a_i=0}$.
{\bf Do.} With $(a_i, y_i, x_i, t_i) \coloneqq (\partial_{\alpha_i}, \partial_{\eta_i}, \partial_{\xi_i}, \partial_{\tau_i})$, compute/implement $\Phi_\ast Z$, with
\[ Z = \omega\exp\left(\sum \lambda_{ij}t_ia_j + \sum q_{ij}y_ix_j +\epsilon P_0\right), \]
$\lambda_{ij}\in\bbZ$, $\omega,q_{ij}\in R\coloneqq \bbQ(T_i=\bbe^{t_i})$, $P_0\in R[a_i,y_i,x_i]$, and
\begin{eqnarray*}
  \Phi^\ast(\bar\alpha_i) &=&
    \sum\psi^1_{ij}\alpha_j + \epsilon P_1, \\
  \Phi^\ast(\bar\eta_i) &=&
    \sum\psi^2_{ij}\eta_j + \epsilon P_2, \\
  \Phi^\ast(\bar\xi_i) &=&
    \sum\psi^3_{ij}\xi_j + \epsilon P_3, \\
  \Phi^\ast(\bar\tau_i) &=&
    \sum\psi^4_{ij}\tau_j + \sum\gamma_{ij}\eta_i\xi_j + \epsilon P_4,
\end{eqnarray*}
$\psi^{1,4}_{ij}\in\bbZ$, $\psi^{2,3}\in R$, $P_{1,4}\in\bbQ[x_i,y_i]$, $P_{2,3}\in R[x_i,y_i]$, $\gamma_{ij}\in R$.

\archived{170713} KZ: $\displaystyle dH = H\sum_{i<j}\frac{dz_i-dz_j}{z_i-z_j}t^{ij}$.

\archived{170610} (alt; main: \refentry{170625}) $\calU_{\hbar;\alpha\beta}$ conventions: $q=\bbe^{\hbar\alpha\beta}$,
$H=\langle a,x\rangle/([a,x]=\alpha x)$ with
\[ A=\bbe^{-\hbar\beta a}, \quad xA=qAx \]
\[ S(a,A,x)=(-a, A^{-1}, -A^{-1}x), \]
\[ \Delta(a,A,x)=(a_1+a_2, A_1A_2, x_1+A_1x_2) \]
and dual $H^\ast=\langle b, y\rangle/([b,y]=-\beta y)$ with
\[ B=\bbe^{-\hbar\alpha b}, \quad By=qyB \]
\[ S(b,B,y)=(-b, B^{-1}, -yB^{-1}), \]
\[ \Delta(b,B,y)=(b_1+b_2, B_1B_2, y_1B_2+y_2). \]
Pairing by $(a,x)^\ast=\hbar(b,y)$ making $\langle a^jx^k,y^lb^j\rangle = \delta_{ij}\delta_{kl}i![k]_q!$.
Then $\calU=H^\ast\otimes H^{op}$ with $(\phi f)(\psi g)=\langle f_1,\psi_1\rangle\langle f_3,S\psi_3\rangle(\phi\psi_2)(gf_2)$.

\archived{170513} (alt; main: \refentry{170625}) $\calU_{\eta,\gamma}$ conventions:
$A=\langle g,G=\bbe^{\eta g},e\rangle/([g,e]=\gamma e)$ with $S(g,G,e)=(-g, G^{-1}, -eG^{-1})$;
\[ \Delta(g,G,e)=(g_1+g_2, G_1G_2, e_1G_2+e_2) \]
and dual $A^\ast=\langle h,H=\bbe^{\gamma h},f\rangle/([h,f]=-\eta h)$ with $S(h,H,f)=(-h, H^{-1}, -H^{-1}f)$;
\[ \Delta(h,H,f)=(h_1+h_2, H_1H_2, f_1+H_1f_2). \]
Pairing by $(g,e)^\ast=(h,f)$. Degrees by $\deg(\gamma,g,e,\eta,h,f)=1$, so ops are degree non-decreasing except the basic pairing lowers 2 degrees.

\archived{170528} (alt; main: \refentry{170625}) $\calU_{\hbar,\epsilon}$ conventions:
$A=\langle g,G=\bbe^{\hbar\epsilon g},e\rangle/([g,e]=\hbar e)$ with $\Delta(g,G,e)=(g_1+g_2, G_1G_2, e_1G_2+e_2)$;  $S(g,G,e)=(-g, G^{-1}, -eG^{-1})$
and dual
$A^\ast=\langle h,H=\bbe^{\hbar h},f\rangle/([h,f]=-\hbar\epsilon h)$ with $\Delta(h,H,f)=(h_1+h_2, H_1H_2, f_1+H_1f_2)$;  $S(h,H,f)=(-h, H^{-1}, -H^{-1}f)$. Pairing by $(g,e)^\ast=(h,f)$. Degrees by $\deg(\hbar,g,e,h,f)=1$, so ops are degree non-decreasing except the basic pairing lowers 2 degrees.

\archived{170412a} {\bf Title.} The Dogma is Wrong. {\bf Abstract.} It has long been known that there are knot invariants associated to semi-simple Lie algebras, and there has long been a dogma as for how to extract them: ``quantize and use representation theory''. We present an alternative and better procedure: ``centrally extend, approximate by solvable, and learn how to re-order exponentials in a universal enveloping algebra''. While equivalent to the old invariants via a complicated process, our invariants are in practice stronger, faster to compute (poly-time vs. exp-time), and clearly carry topological information.
\P\ This is joint work with Roland van der Veen and continues work by Rozansky and Overbay.

% Revised from http://tex.stackexchange.com/questions/12895/overlay-symbol-with-another:
%\makeatletter
%\def\@Ou{{\setbox0\hbox{$\bigcirc$}\rlap{\hbox to \wd0{\hss u\hss}}\box0}}
%\def\Ou{\mathrel{\mathchoice{\@Ou}{\@Ou}{\scriptsize\@Ou}{\tiny\@Ou}}}
%\makeatother
% From http://tex.stackexchange.com/questions/249688/how-do-i-draw-a-circle-around-a-term-in-an-align-equation:
%\makeatletter
%\newcommand\mathcircled[1]{%
%  \mathpalette\@mathcircled{#1}%
%}
%\newcommand\@mathcircled[2]{%
%  \tikz[baseline=(math.base)] \node[draw,circle,inner sep=1pt] (math) {$\m@th#1#2$};%
%}
%\makeatother
%\def\OOu{\scalebox{0.5}{$\mathcircled{\bigcup}$}}

%\needspace{7.8cm}
\parshape 9 0in \linewidth 0in \linewidth 0in \linewidth 0in 1.8in 0in 1.65in 0in 1.5in 0in 1.35in 0in 1.2in 0in 1.0in
\archived{170401o} {\bf Project} over-then-under ``$\OU$-Tangles''. Closed under compositions; (v-)braids are $\OU$; non-braid $\OU$ tangles? Relations in $\OU$? In $\calA^\OU$? Not all tangles are $\OU$. Alexander properties; v-version. Associators in $\calA^u\cap\calA^\OU$: Constructible? Sufficient for EK? Relations with Chterental's ``virtual curve diagrams''? Chu's syzygy:
\newline\vskip -6.7\baselineskip
\includegraphics[width=\linewidth]{figs/SyzygyByChu.png}

\archived{170108b} \href{http://drorbn.net/index.php?title=AKT-17}{AKT-17} Reality $\act$ Plan: {\bf Gentle.} Course introduction (h1). Knots, Reidemeister moves and the Jones polynomial (h2-3). Tangles and a faster Jones program (h4). Tangles and meta-monoids (h5-6). Links, 3-manifolds, Seifert surfaces and genus, ribbon knots and ``algebraic knot theory'' (h7-8). The Alexander polynomial using $\Gamma$-calculus (h9-10). Finite type invariants and expansions (h11-14). $/$ The relationship with metrized Lie algebras and PBW (h15-16). $/$ The variants $v$, $w$, $bv$, and $rv$, and their expansions (h17-18). Lie bialgebras and solvable approximation (h19-20).
{\bf Brute.} Knots, algebras, YBE, CYBE, Lie algebras, universal enveloping algebras, formulas (h1) The Lie algebra $\frakg_0$, universal enveloping algebras and low degree computations (h2-3). Ordering symbols and commutation relations for $\frakg_0$ (h4-5). The $\frakg_0$ invariant (h6-7). The {\greektext L'ogos} and $\frakg_1$ computations (h8-10). $\act$ Morse knots and the $\frakg_1$ invariant (h11). $\frakg_0$ and $\frakg_1$ as approximations of $sl_2$, approximating $sl_3$ (h12). The $sl_3^0$ invariant (h13-14). The $sl_3^1$ invariant, fame, and glory (h15-16).

\archived{160513} {\bf Q.} What's Fox-Milnor for links? \refentry{131130b}.

\archived{170211a} Gaussian pairing:
\newline$\left\langle
  \exp\left(\frac{x\subset}{2}+\sum_{i\in I}i\!\multimapdotinv\right)
  \mid
  \exp\left(\frac{\supset y}{2}+\sum_{j\in J}\multimapdot\!j\right)
\right\rangle =$
\newline
\null\hfill$\exp\left(\!\log(\frac{1}{1-xy})\bigcirc
  \!+\! \sum_{i\in I, j\in J}\frac{i\multimapdotboth j}{1-xy}
  \!+\! \sum_{i_{1,2}\in I}\frac{\prescript{i_1}{i_2}\multimapdotbothvert y}{1-xy}
  \!+\! \sum_{j_{1,2}\in J}\frac{x\multimapdotbothvert^{j_1}_{j_2}}{1-xy}
\right)$.

\archived{160314} {\bf Proj.} Visualization of fibred knots. [Done, \pdftooltip{{\red JB}}{Jesse Bettencourt's project}\href{http://jessebett.com/TorusKnotFibration/}{$\to$}].

\archived{160403a} Is $\Box$ on unipotent completions (page 7 of my GT1 paper) nonsense? Are Taylor expansions isomorphisms?

\archived{160321} {\bf Prob.} Find a quadratic description for the Adjoint rep of $[\bar{a}_{ik},\bar{a}_{jk}] = -[\bar{a}_{ij},\bar{a}_{jk}] = \bar{a}_{ik}-\bar{a}_{jk}$,
$[\bar{a}_{ij},\bar{a}_{ji}] = \bar{a}_{ji}-\bar{a}_{ij}$.

\archived{141123} Qinhuangdao: Talk to me about China, America, Taiwan, economy, ecology, religion, democracy, censorship, and all else.

\archived{150201b} A precise relationship between expansions for $\FG$ and ``PBW bases'' for same?

\archived{150401} Halacheva's meta-trace: $\displaystyle
\begin{CD}
  \begin{array}{c|ccc}
    \omega & c & S \\
    \hline
    c & \alpha & \theta \\
    S & \psi & \Xi
  \end{array}
  \hspace{-2.2mm}@>\Gamma::\tr_c>\mu\coloneqq 1-\alpha>\hspace{-1.8mm}
  \begin{array}{c|cc}
    \mu\omega & S \\
    \hline
    S & \!\Xi+\psi\theta/\mu\!
  \end{array}
\end{CD}$
\vskip -3mm
with $\displaystyle\begin{CD}
  \begin{array}{c|ccc}
    \omega & a & b & S \\
    \hline
    a & \alpha & \beta & \theta \\
    b & \gamma & \delta & \epsilon \\
    S & \phi & \psi & \Xi
  \end{array}
  @>\Gamma::m^{ab}_c>{\mu\coloneqq 1-\beta \atop T_a,T_b\to T_c}>
  \begin{array}{c|cc}
    \mu\omega & c & S \\
    \hline
    c & \gamma+\alpha\delta/\mu & \epsilon+\delta\theta/\mu \\
    S & \phi+\alpha\psi/\mu & \Xi+\psi\theta/\mu
  \end{array}
\end{CD}$.
When exactly is it defined?

\archived{141211} \bbs{Alekseev}{131108}{071125}, AT sec.\ 5.2: $F_1\in\text{TAut}$ solving AT $\leftrightarrow$ $F_t\coloneqq F_1(tx,ty)$ with $F_0=1$ $\leftrightarrow$ $u_t=\frac{dF_t}{dt}F_t^{-1}$ with $u_t=\frac{1}{t}u(tx,ty)$ $\leftrightarrow$ $\tder\ni u=(A,B)$ solving KV. What means $F_1(tx,ty)$?

\archived{140228a} {\bf Proj.} Associator computations using {\tt FreeLie`}.

\archived{140203} {\bf Project} ``expansions and quadraticity for groups'': definitions, relations with Hain / Mal'cev / Quillen / Vassiliev, torsion, semi-direct products, $FG$, $\PuB$, $\PvB$, $\PwB$ (and homotopy versions), elliptic / higher genus braids, mapping class groups, right-angled Artin groups, Stallings' theorem, knot groups (u, w, higher D), Hutchings-Lee. ?`Flat braids after Merkov, Hilden braids, $[\PuB_n,\PuB_n]$ (also $u\to v,w$), $[G,G]$ in general, $\Aut(\FG_2)$, $\Aut(\FG_n)$, Torelli following Hain?

\archived{150107} {\bf Paperlet.} ``An algebraic characterization of the Taylor expansion''.

\archived{141209}\newline\includegraphics[width=\columnwidth]{figs/CirclePair4T.pdf}

\archived{141129} Implement \verb$SeriesSolve$:

\includegraphics[width=\columnwidth]{figs/SeriesSolve.pdf}

\archived{140627} {\bf Proj.} A 3-page paper on Gassner and its unitarity.

\archived{140119} Lie-series Mathematica abstraction challenge (also {\tt 2014-01}): \yellowt{\tt LieSeries}, \yellowt{\tt MakeLieSeries}, {\tt Crop}, {\tt RandomLieSeries}, \yellowm{+}, $c\cdot$, \yellowm{\equiv}, \yellowm{\int}, \yellowm{b}, {\tt EulerE}, {\tt adPower}, \yellowt{\tt adSeries}, {\tt Ad}, {\tt LieDerivation} (\yellowt{also on {\tt CW}}, {\tt AW}), $+$, $c\cdot$, {\tt DerivationPower}, {\tt DerivationSeries}, {\tt LieMorphism} (also on {\tt CW}, {\tt AW}, $\langle\rangle$ and \yellowt{into $\langle\rangle$}), {\tt StableApply}, \yellowt{\tt BCH}, {\tt ASeries}, {\tt MakeASeries}, $\iota$, $\sigma$, {\tt CWSeries}, {\tt MakeCWSeries}, {\tt RandomCWSeries}, $+$, $c\cdot$, $\equiv$, $\int$, $\attr$, $\atdiv$ \yellowt{\tt JA}, \yellowm{\langle\dots\rangle}, $+$, $c\cdot$, {\tt TangentialDerivation}, \yellowm{tb}, \yellowm{\Gamma}, \yellowm{\Gamma^{-1}}.

\archived{140213} Brochier's even associators to degree 9 at \url{http://abrochier.org/sage.php}.

\archived{140112} Carter-Saito: moves on decker curves:
\par\includegraphics[width=0.515\columnwidth]{../../2013-07/Vis4D/Carter/SurfacesBook-p42.png}%
\hfill\includegraphics[width=0.475\columnwidth]{../../2014-01/CarterSaito-p134.png}

\archived{140113} BF perturbation theory in ambient axial gauge ($A$-$B$ propagator is $t$-vertical, $B$ above $A$; inner gauge unspecified):
\[ \includegraphics[width=0.8\columnwidth]{../../2014-01/BFFeynmanRules.png} \]

\parbox[b]{1.05in}{{\bf\tiny (131202c)} \raggedright At {\tt 2013-12/ 4DHardware/}:}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-01.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-02.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-03.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-04.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-05.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-06.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-07.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-08.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-09.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-10.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-11.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-12.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-13.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-14.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-15.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-16.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-17.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-18.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-19.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-20.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-21.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-22.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-23.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-24.png}
\hfill\includegraphics[height=0.66in]{../../2013-12/4DHardware/4DHardware-25.png}

\archived{140119} At {\tt 2014-01}, an abstraction challenge:
\par\includegraphics[width=\columnwidth]{../../2014-01/AbstractionChallenge-1.png}
\par\includegraphics[width=\columnwidth]{../../2014-01/AbstractionChallenge-2.png}

\archived{131111} Missing in {\tt FreeLie.m}: The relation with $\tder_n$:
$\exp\left(\sum_i\ad_{u_i}\{\gamma_i\}\right) = C_{u_1,u_2,\ldots}^{\beta_1,\beta_2,\ldots}$, etc.

\archived{140109} Virtual 2-knots:
\[ \graphicspath{{../../2014-01/}} \input{../../2014-01/Virtual2Knots.pdf_t} \]

\archived{131229} What are the two winding numbers for immersions $\bbR^2\hookrightarrow\bbR^4$? Is every pair realized? Is there a Whitney-Graustein theorem?

\archived{131130a} Meilhan: Levine: \arXiv{q-alg/9711007} {\em A Factorization of the Conway Polynomial}. Then Tsukamoto, Yasuhara: \arXiv{math/0405481} {\em A factorization of the Conway polynomial and covering linkage invariants}.

\archived{131026} Time to make an ``agenda browser''.

\archived{131017} If $\lambda_{\{ij\}}=0$,
$(\lambda_{ij}dx^i\wedge dx^j)^{n/2} = \sqrt{\det(\lambda_{ij})}\bigwedge_idx^i$.

\end{multicols}

\end{document}

\endinput
