<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[11pt,notitlepage]{article}
\def\bare{n}
\usepackage[all]{xy}
\usepackage[english,greek]{babel}
\usepackage{dbnsymb, amsmath, graphicx, amssymb, multicol, stmaryrd, pifont,
  amscd, colortbl, mathtools, wasysym, needspace, import, longtable, overpic,
  enumitem, bbm, pdfpages, ../picins, array, setspace, datetime}
\usepackage[export]{adjustbox} % Follows https://tex.stackexchange.com/questions/6073/scale-resize-large-images-graphics-that-exceed-page-margins
\usepackage{tensor}
\usepackage{txfonts}	% for the likes of \coloneqq.
\usepackage{fontawesome} % for \faPlay
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{utfsym}	% for the likes of \car=\usym{1F697}
\usepackage{soul} % for strikeouts, \st.
\usepackage[textwidth=8.5in,textheight=11in,centering]{geometry}
\parindent 0in
\usepackage[makeroom]{cancel}

% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}
\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor={blue!50!black}
}

% Following http://tex.stackexchange.com/questions/59340/how-to-highlight-an-entire-paragraph
\usepackage[framemethod=tikz]{mdframed}

\usepackage[T1]{fontenc}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Pitzer-250308}
\def\title{A Seifert Dream}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to Pitzer College!}}
\def\webdef{{{\greektext web}$\coloneqq$\href{http://drorbn.net/pi25}{http://drorbn.net/pi25}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}

\definecolor{mblue}{HTML}{E0E0FF}
\definecolor{mgray}{HTML}{B0B0B0}
\definecolor{mgreen}{HTML}{00B000}
\definecolor{morange}{HTML}{FFA50A}
\definecolor{mpink}{HTML}{FFE0E0}
\definecolor{mpurple}{HTML}{B000B0}
\definecolor{myellow}{HTML}{FFFF00}
\def\blue{\color{blue}}
\def\gray{\color{gray}}
\def\mgray{\color{mgray}}
\def\mgreen{\color{mgreen}}
\def\morange{\color{morange}}
\def\pink{\color{pink}}
\def\mpurple{\color{mpurple}}
\def\magenta{\color{magenta}}
\def\red{\color{red}}
\def\yellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{yellow}{$#1$}}}
\def\myellowm#1{{\setlength{\fboxsep}{0pt}\colorbox{myellow}{$#1$}}}
\def\mpinkm#1{{\setlength{\fboxsep}{0pt}\colorbox{mpink}{$#1$}}}
\def\mbluem#1{{\setlength{\fboxsep}{0pt}\colorbox{mblue}{$#1$}}}
\def\cbox#1#2{{\setlength{\fboxsep}{0pt}\colorbox{#1}{#2}}}

\def\arXiv#1{{\href{http://arxiv.org/abs/#1}{{\tiny arXiv:}\linebreak[0]{#1}}}}

\def\qed{{\linebreak[1]\null\hfill\text{$\Box$}}}

\def\act{{\hspace{-1pt}\sslash\hspace{-0.75pt}}}
\def\ad{\operatorname{ad}}
\def\Ad{\operatorname{Ad}}
\def\aft{$\overrightarrow{\text{4T}}$}
\def\AS{\mathit{AS}}
\def\bbZZ{{\mathbb Z\mathbb Z}}
\def\CW{\text{\it CW}}
\def\diag{\operatorname{diag}}
\def\eps{\epsilon}
\def\FL{\text{\it FL}}
\def\Hom{\operatorname{Hom}}
\def\IHX{\mathit{IHX}}
\def\mor{\operatorname{mor}}
\def\PvT{{\mathit P\!v\!T}}
\def\remove{\!\setminus\!}
\def\STU{\mathit{STU}}
\def\SW{\text{\it SW}}
\def\TC{\mathit{TC}}
\def\tr{\operatorname{tr}}
\def\vT{{\mathit v\!T}}

\def\bara{{\bar a}}
\def\barb{{\bar b}}
\def\barT{{\bar T}}
\def\bbE{{\mathbb E}}
\def\bbe{\mathbbm{e}}
\def\bbH{{\mathbb H}}
\def\bbN{{\mathbb N}}
\def\bbO{{\mathbb O}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbZ{{\mathbb Z}}
\def\bcA{{\bar{\mathcal A}}}
\def\calA{{\mathcal A}}
\def\calD{{\mathcal D}}
\def\calF{{\mathcal F}}
\def\calG{{\mathcal G}}
\def\calH{{\mathcal H}}
\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\fraka{{\mathfrak a}}
\def\frakb{{\mathfrak b}}
\def\frakg{{\mathfrak g}}
\def\frakh{{\mathfrak h}}
\def\tilE{\tilde{E}}
\def\tilq{\tilde{q}}

\def\tDelta{\tilde{\Delta}}
\def\tf{\tilde{f}}
\def\tF{\tilde{F}}
\def\tg{\tilde{g}}
\def\tI{\tilde{I}}
\def\tm{\tilde{m}}
\def\tR{\tilde{R}}
\def\tsigma{\tilde{\sigma}}
\def\tS{\tilde{S}}
\def\tSW{\widetilde{\SW}}

\def\car{\reflectbox{\usym{1F697}}}
\def\rac{\usym{1F697}}

% From http://tex.stackexchange.com/questions/154672/how-to-get-a-medium-sized-otimes
\DeclareMathOperator*{\midotimes}{\text{\raisebox{0.25ex}{\scalebox{0.8}{$\bigotimes$}}}}

% \Gint from https://tex.stackexchange.com/questions/171415/superimpose-letter-on-integral-symbol
\makeatletter
\let\DOTSI\relax % amsmath support for \dots
\newcommand*{\Gint}{%
  \DOTSI
  \mathop{%
    \mathpalette\@LetterOnInt{G}%
  }%
  \mkern-\thinmuskip % thin space is inserted between two \mathop
  \int
}
\newcommand*{\@LetterOnInt}[2]{%
  \sbox0{$#1\int\m@th$}%
  \sbox2{$%
    \ifx#1\displaystyle
      \textstyle
    \else
      \scriptscriptstyle
    \fi
    #2%
  \m@th$}%
  \dimen@=.4\dimexpr\ht0+\dp0\relax
  \ifdim\dimexpr\ht2+\dp2\relax&gt;\dimen@
    \sbox2{\resizebox*{!}{\dimen@}{\unhcopy2}}%
  \fi
  \dimen@=\wd0 %
  \ifdim\wd2&gt;\dimen@
    \dimen@=\wd2 %
  \fi
  \rlap{\hbox to \dimen@{\hfil
    $#1\vcenter{\copy2}\m@th$%
  \hfil}}%
  \ifdim\dimen@&gt;\wd0 %
    \kern.5\dimexpr\dimen@-\wd0\relax
  \fi
}
\makeatother

%%%

\def\Abstract{{\raisebox{1.2mm}{\parbox[t]{3.95in}{
{\red\bf Abstract.}  Given a knot $K$ with a Seifert surface $\Sigma$,
I dream that the well-known Seifert linking form $Q$, a quadratic form on
$H_1(\Sigma)$, has plenty docile local perturbations $P_\eps$ such that
the formal Gaussian integrals of $\exp(Q+P_\eps)$ are invariants of $K$.
\par In my talk I will explain what the above means, why this dream is
oh so sweet, and why it is in fact closer to a plan than to a delusion.
\hfill Joint with Roland van der Veen.
}}}}

\def\SA{{\raisebox{1.2mm}{\parbox[t]{2.65in}{
{\red\bf The Seifert-Alexander Formula.} With $P,Q\in H_1(\Sigma)$,
\[ Q(P,G) = T^{1/2}lk(P^+,G) - T^{-1/2}lk(P,G^+) \]
\[ \Delta(K) = \det(Q) \]
\[ \int_{2H_1(\Sigma)}\hspace{-3mm}dp\,dx\,\exp Q(p,x) \doteq \det(Q)^{-1} \]
(where $\doteq$ means ``ignoring silly factors'').
}}}}

\def\Feynman{{\raisebox{1.2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 2.65in 0in 2.65in 0in 2.65in 0in 2.65in 0in 3.95in
{\red\bf Perturbed Gaussian Integration.} We say that
$P_\eps\in\eps\bbQ[x_1,\ldots x_n]\llbracket\eps\rrbracket$ is $M$-docile
(for some $M\colon\bbN\to\bbN$) if for every monomial $m$ in $P_\eps$
we have $\deg_{x_1,\ldots,x_n}(m)\leq M(\deg_\eps(m))$.

{\red\bf Theorem} (Feynman). If $Q$ is a quadratic in $x_1,\ldots,x_n$ and $P_\eps$ is
docile, set $Z_\eps = \int_{\bbR^n}dx_1\cdots x_n\,\exp\left(Q+P_\eps\right)$.
Then every coefficient in the $\eps$-expansion of $Z_\eps$ is computable in polynomial time
in $n$. in fact,
\vskip 2mm
\par\ $\Delta^{1/2}Z_\eps \doteq 
  \left\langle\exp Q^{-1}(\partial_{x_i}), \exp P_\eps\right\rangle = $

\vskip 8mm
{\red $\theta(T,1)$ is like that!} With $\eps^2=0$,
}}}}

\def\ds{\displaystyle}
\def\vs#1{$\bbR^2_{p_#1x_#1}$}
\def\rp#1#2{$\calL(X^+_{#1#2})$}
\def\gp#1#2{$\calL(C^{#1}_#2)$}
\def\ta#1{$\tau(p_#1,x_#1)$}
\def\fintexample{$\ds Z \doteq \Gint_{\mathrlap{2E=\bbR^{14}_{p_ix_i}}}
  \ \ {\red\calL(X^+_{15})} {\mgreen\calL(X^+_{62})} {\blue\calL(X^+_{37})}
{\mpurple\calL(C^{-1}_4)}
$}

\def\sltwodefs{{\raisebox{0mm}{\parbox[t]{2.6in}{\setstretch{1.25}
where $\calL(X^s_{ij}) \doteq \bbe^{L(X^s_{ij})}$,
$\calL(C^\varphi_i) \doteq \bbe^{L(C^\varphi_i)}$, 
\par $\ds L(X^s_{ij}) = x_i(p_{i+1}-p_i) + x_j(p_{j+1}-p_j)$
\par\hfill $\ds + (T^s-1)x_i(p_{i+1}-p_{j+1})$
\vskip 3pt\par\hfill $\ds +  \frac{\eps s}{2} \left(
    x_i (p_i-p_j) \left({(T^s-1)x_ip_j\quad}\atop{\quad+2(1-x_jp_j)}\right)-1
  \right)$
\vskip 5pt\par$\ds L(C^\varphi_i) = x_i(p_{i+1}-p_i) + \eps\varphi(1/2-x_ip_i)$
\par{\red $\theta(T_1,T_2)$ is likewise,} with harder formulas and integration
over $6E$.
}}}}

\def\Dream{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Dream.} There is a similar perturbed Gaussian integral formula
for $\theta$, but with integration over $6H_1(\Sigma)$. The quadratic
$Q$ will be the same as in the Seifert-Alexander formula (but repeated
3 times, for each $T_\nu$). The perturbation $P_\eps$ will be given
by low-degree finite type invariants of curves on $\Sigma$ (possibly
also dependent on the intersection points of such curves, or on other
information coming from $\Sigma$).

{\red\bf Evidence.} Experimentally (yet undeniably), $\deg\theta$
is bounded by the genus of $\Sigma$. How else could such a genus bound
arise? Further very strong evidence comes from the conjectural (yet undeniable) understanding
of $\theta$ as the two-loop contribution to the Kontsevich integral
\cite{Ohtsuki:TwoLoop} and/or as the ``solvable approximation'' of the
universal $sl_3$ invariant \cite{DPG, PG}.

%\parshape 1 0in 3in
{\red\bf Why so sweet?} It will allow us to prove the aforementioned
genus bound and likely, the hexagonal symmetry. Sweeter and dreamier,
it may allow us to say something about ribbon knots!
}}}}

\def\Local{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf What's ``local''? How will we compute?} The B\k{e}dlewo Alexander formula: Let $F$
be the faces of a knot diagram. Make an $F\times F$ matrix $A$ by adding for each crossing 
contributions
\[
  {\arraycolsep=0pt \begin{array}{ccc}
    &amp;k&amp;\\[-8pt]
    l&amp;\raisebox{-4pt}{\scalebox{3}{$\overcrossing$}}&amp;j\\[-6pt]
    &amp;i&amp;
  \end{array}}
  \to {\arraycolsep=2pt \begin{pmatrix}
    -1 &amp; -1 &amp; 2  &amp; 0 \\
    0  &amp; 0  &amp; 0  &amp; 0 \\
    0  &amp; 1  &amp; -1 &amp; 0 \\
    1  &amp; 0  &amp; -1 &amp; 0
  \end{pmatrix}}
  \qquad
  {\arraycolsep=0pt \begin{array}{ccc}
    &amp;k&amp;\\[-8pt]
    l&amp;\raisebox{-4pt}{\scalebox{3}{$\overcrossing$}}&amp;j\\[-6pt]
    &amp;i&amp;
  \end{array}}
  \to {\arraycolsep=2pt \begin{pmatrix}
    1  &amp; -1 &amp; 0  &amp; 0 \\
    0  &amp; 0  &amp; 0  &amp; 0 \\
    -2 &amp; 1  &amp; 1  &amp; 0 \\
    1  &amp; 0  &amp; -1 &amp; 0
  \end{pmatrix}}
\]
at rows / columns $(i,j,k,l)$. Then $\Delta =
\det'\left((T^{1/2}A-T^{-1/2}A)/2\right)$.
\vskip -3.5mm
\[
  \includegraphics[width=0.8in, valign=c]{SeifertByRedelmeier-1.jpg}
  \to\includegraphics[width=0.8in, valign=c]{SeifertByRedelmeier-2.jpg}
  \to\includegraphics[width=0.8in, valign=c]{SeifertByRedelmeier-3.jpg}
  \to\includegraphics[width=0.8in, valign=c]{SeifertByRedelmeier-4.jpg}
\]
\vskip -2mm
{\tiny (the Seifert algorithm by Emily Redelmeier)}

\vskip -1mm
{\red Expect the like for $\theta$! Expect more like $\theta$! Topology first! Resist the
tyranny of quantum algebra!}
}}}}

\def\Captions{{\raisebox{1.2mm}{\parbox[t]{3.95in}{
{\bf\red Right.} The 132-crossing torus knot $T_{22/7}$ (more at
\web{TK}).  \par{\bf\red Below.} Random knots from \cite{DHOEBL:Random},
with 101-115 crossings (more at \web{DK}).
}}}}

\pagestyle{empty}

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

\newsavebox\bib
\savebox\bib{\parbox{\textwidth}{
  \def\bysame{{---}}
  \renewcommand{\section}[2]{}%
  \begin{thebibliography}{}
  \import{../UBC-241004}{refs.tex}
  \end{thebibliography}
}}

\begin{center} \null\vfill\input{ASDT.pdftex_t}\vfill\null \end{center}

\includepdf[pages={1-3}]{SGC24.pdf}

\end{document}

\endinput

</pre></body></html>