\documentclass[11pt,notitlepage]{article} \usepackage{amsmath, amssymb} \usepackage{graphicx, stmaryrd, datetime, amscd, multicol, txfonts, ../picins, mathabx, dbnsymb, tensor, needspace, ulem, mathtools, enumitem} \usepackage{mathbbol} % 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, %\usepackage{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 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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}}} \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}, 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\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\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\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\PvT{\text{\it P\!v\!T}} \def\PwB{\text{\it P\!w\!B}} \def\bb1{{\mathbb 1}} \def\bbA{{\mathbb A}} \def\bbB{{\mathbb B}} \def\bbC{{\mathbb C}} \def\bbD{{\mathbb D}} \def\bbe{\mathbb{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\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\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{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}. \entry{210410} Merz @ \arXiv{2104.02993} discusses formulas for the additivity defects of signatures of braid closures. \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? \entry{210302} {\bf Q.} (w/ Abbasi) Does every contractible but not manifestly contractible curve in the annulus have an odd self-intersection? \needspace{0mm} \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. \entry{200703a} {\bf Q.} An Archibald calculus that includes strand doubling? A common generalization of Archibald- and $\Gamma$-calculus? \refentry{200804}, \refentry{210211a} \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}\ $? Canonical forms for morphisms? Carries Alexander? \href{http://drorbn.net/AcademicPensieve/2021-01/nb/Det3x3.pdf}{2021-01: Det3x3.nb}, \refentry{181222b}, ``weighted automata'', ``rational series''. \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 $\epsilon=1$. \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{210101} {\bf Proj.} Implement a {\tt PD2ThinMorse} with guaranteed bounds and/or using simulated annealing. \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{1mm} % 0mm 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 $\left.S_{ab}\right|_{e=t^{-1}}=R^{-1}_{ba}$. \entry{201225b} {\bf Proj.} Develop poly-dimensional braid representations via ``Burau/Gassner homology''. \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? \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 $\calU(\frakg)_\frakg$. \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} \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{200611a} {\bf Q.} What conditions on $(\bbA,\bbB,\langle\cdot\rangle)$ are enough to make the Drinfel'd double associative? Examples beyond Hopf algebras? \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 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''? {\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{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? Following \href{http://drorbn.net/dbnvp/M19_Ito.php}{Ito@M19}, relations with Garside lengths? \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 $U\oplus V\oplus\mathbb{1}$: \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_{iF(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 $H\coloneqq F(M^2)$: \[ \Delta_H\colon\ \begin{CD} F(M^2)@>F(\Delta\Delta)>>F(M^4)@>F(1R1)>>F(M^4)@>c_{M,M}>>F(M^2)^2, \end{CD} \] \[ m_H\colon\qquad \begin{CD} F(M^2)^2 @F(1\epsilon 1)>> F(M^2), \end{CD} \] \[ S_H\colon\qquad\qquad \begin{CD} F(M^2) @>F(R)>> F(M^2). \end{CD} \] 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{1mm} % 0mm 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{190108} {\bf Do.} With $P=\sum a_{mn}z^m\zeta^n$, compute $\llangle\epsilon P\rrangle \coloneqq \log\langle\exp\epsilon P\rangle$. \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 wn(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 $bi}\alpha_i}$, $\sigma = \frac{1-T}{\hbar}\sum_{i}[d] \\ \calA^v(\uparrow\uparrow^{ab}) \ar[r]^<>(0.5){T_{\frakg^+}} & \calU(\frakg^+) }$} \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{170316} \href{https://en.wikipedia.org/wiki/Wigner-Weyl_transform}{Wikipedia: Wigner-Weyl transform}: Gaussians compose hyperbolically, $\exp \left (-{a } (q^2+p^2)\right )\star \exp \left (-{b} (q^2+p^2)\right )$ \newline\null\hfill$= {1\over 1+\hbar^2 ab} \exp \left (-{a+b\over 1+\hbar^2 ab} (q^2+p^2)\right )$. \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 \href{http://mathoverflow.net/questions/7052/}{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{2mm} % 1mm 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{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? \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 $0}$ 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{150416} Chterental: Is there a Melvin-Morton statement for v-knots? \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{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. \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. $H[r]^i & (M,b) \ar@<2pt>[l]^p}$, 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{0cm} \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{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 $h_i=h'_i-\epsilon g_i$, $[h_i,\cdot]=0$, $[g_i,g_j]=0$, $[e_{ij},e_{kl}]=\delta_{jk}e_{il}-\delta_{il}e_{kj}$, $[f_{ij},f_{kl}]=\epsilon(\delta_{jk}f_{il}-\delta_{il}f_{kj})$, $[e_{ij},f_{kl}] = \delta_{jk}(\epsilon\delta_{il}f_{il}) - \delta_{li}(\epsilon\delta_{kj}f_{kj}) + \delta_{jk}\delta_{li}((h_i-h_j)/2+\epsilon(g_i-g_j))$, $[g_i,e_{jk}]=(\delta_{ij}-\delta_{ki})e_{jk}$, $[g_i,f_{jk}]=(\delta_{ij}-\delta_{ki})f_{jk}$, $\deg(\epsilon,h_i,f_{ij},g_i,e_{ij})=(1,1,1,0,0)$. 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: $\displaystyle dH = H\sum_{i\Gamma::\tr_c>\mu\coloneqq 1-\alpha>\hspace{-1.8mm} \begin{array}{c|cc} \mu\omega & S \\ \hline S & \!\Xi+\psi\theta/\mu\! \end{array} \end{CD}$ \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