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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Geneva-2208}
\def\title{Tangles in a Pole Dance Studio: A Reading of Massuyeau, Alekseev, and Naef}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to Geneva!}}
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%%%

\def\PrelDefs{{\raisebox{2mm}{\parbox[t]{3.125in}{
{\red\bf Preliminary Definitions.} Fix $p\in\bbN$ and $\bbF=\bbQ/\bbC$.
Let $D_p\coloneqq D^2\setminus(p\text{ pts})$, and let the {\red Pole
Dance Studio} be $\PDS_p\coloneqq D_p\times I$.
}}}}

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 5 0in 2.25in 0in 2.25in 0in 2.25in 0in 2.25in 0in \linewidth
{\red\bf Abstract.} I will report on joint \text{work} with Zsuzsanna
\text{Dancso}, \text{Tamara} \text{Hogan}, Jessica Liu, and Nancy Scherich. Little
of what we do is original, and much of it is simply a reading of
\text{Massuyeau} \cite{Massuyeau:FormalDescription} and Alekseev
and \text{Naef}~\cite{AlekseevNaef:GTfromKZ}.

\parshape 8 0in 2.05in 0in 2.05in 0in 2.05in 0in 2.05in 0in 2.05in 0in 2.05in 0in 2.05in 0in \linewidth
We study the pole-strand and strand-strand double filtration on the space
of tangles in a pole dance studio (a punctured disk cross an interval),
the corresponding homomorphic expansions, and a strand-only HOMFLY-PT
relation. When the strands are transparent or nearly transparent to
each other we recover and perhaps simplify substantial parts of the
work of the aforementioned authors on expansions for the Goldman-Turaev
Lie bi-algebra.\hfill{\red $\Longrightarrow$}~\raisebox{2pt}{\resizebox{!}{5pt}{\parbox{3in}{
Expansions $W\colon\FG\langle X_i\rangle\to\FA\langle x_i\rangle$:
\newline Magnus: $X_i\mapsto 1+x_i$, $X_i^{-1}\mapsto 1-x_i+x_i^2-\ldots$
\newline Exponential: $X_i^{\pm 1}\mapsto\bbe^{\pm x_i}$
}}}
}}}}

\def\MainThm{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Definitions.}
Let $\pi\coloneqq\FG\langle X_1,\ldots,X_p\rangle$ be the free group
(of deformation classes of based curves in $D_p$), $\bar\pi$ be the
framed free group (deformation classes of based immersed curves),
$|\pi|$ and $|\bar\pi|$ denote $\bbF$-linear combinations of
cyclic words ($|x_iw|=|wx_i|$, unbased curves), $A\coloneqq\FA\langle
x_1,\ldots,x_p\rangle$ be the free associative algebra, and let
$|A|\coloneqq A/(x_iw=wx_i)$ denote cyclic algebra words.

\vskip 0.6in
{\red\bf Theorem 1} (Goldman, Turaev, Massuyeau, Alekseev, Kawazumi, Kuno, Naef). $|\bar\pi|$ and
$|A|$ are Lie bialgebras, and there is a ``homomorphic expansion''
$W\colon|\bar\pi|\to|A|$: a morphism of Lie bialgebras with $W(|X_i|)=1+|x_i|+\ldots$.
}}}}

\def\FDefs{{\raisebox{2mm}{\parbox[t]{3.95in}{
\parshape 1 0in 3.3in
{\bf\red Further Definitions.}
$\bullet$\ $\calK=\calK_0=\calK_0^0=\calK(S) \coloneqq$
\newline\null\hfill$\bbF\langle\text{framed tangles in }\PDS_p\rangle$.

\parshape 2 0in 3.3in 0in \linewidth
$\bullet$\ $\calK_t^s\coloneqq$(the image via
$\doublepoint\to\overcrossing-\undercrossing$ of tangles in $\PDS_p$
\newline\null\hfill that have $t$ double points, of which $s$ are strand-strand).
\newline\null\quad E.g.,

\vskip 8mm
$\bullet$\ $\calK^{/s}\coloneqq\calK/\calK^s$.\quad Most important, {\red $\calK^{/1}(\bigcirc)=|\bar\pi|$}, and there is
\newline\null\hfill $P\colon\calK(\bigcirc)\to|\bar\pi|$.

$\bullet$\ $\calA\coloneqq\prod\calK_t/\calK_{t+1}$,\quad
  $\calA^s\coloneqq\prod\calK^s_t/\calK^s_{t+1} \subset \calA$,\quad
  $\calA^{/s}\coloneqq\calA/\calA^s$.
}}}}

\def\Facts{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Fact 1.} The Kontsevich Integral is an ``expansion''
$Z\colon\calK\to\calA$, compatible with several noteworthy structures.

{\red\bf Fact 2} (Le-Murakami, \cite{LeMurakami:HOMFLY}). $Z$ satisfies the strand-strand HOMFLY-PT
relations: It descends to $Z_H\colon\calK_H\to\calA_H$, where
\[ \calK_H\coloneqq\calK\left/\left(
    \text{\red\Large$\overcrossing$} - \text{\red\Large$\undercrossing$}
      = (\bbe^{\hbar/2}-\bbe^{-\hbar/2})\cdot\text{\red\Large$\upupsmoothing$}
  \right)\right.
\]
\[ \calA_H\coloneqq\calA\left/\left(
    \qquad=\hbar\qquad\text{ or }\qquad=\hbar\qquad
  \right)\right.
\]
and $\deg\hbar=(1,1)$.

{\red\bf Proof of Fact 2.} $\displaystyle Z(\overcrossing)-Z(\undercrossing)
  = \virtualcrossing\cdot\left(\bbe^{\horizontalchord/2}-\bbe^{-\horizontalchord/2}\right)
$
\newline\null\hfill$\displaystyle
  = \virtualcrossing\cdot\left(\bbe^{\hbar\virtualcrossing/2}-\bbe^{-\hbar\virtualcrossing/2}\right)
  = \left(\bbe^{\hbar/2}-\bbe^{-\hbar/2}\right)\upupsmoothing.
  \quad\qed$
}}}}

\def\onleft{{$\calK_5^2(\bigcirc)=$}}
\def\onright{{$/.\ \ \doublepoint\to\overcrossing-\undercrossing$}}

\def\Keys{{\raisebox{2.5mm}{\parbox[t]{3.95in}{
{\red\bf Key 1.} $W\colon|\bar\pi|\to|A|$ is $Z^{/1}_H\colon\calK^{/1}_H(\bigcirc)\to\calA^{/1}_H(\bigcirc)$.

{\red\bf Key 2} (Schematic). Suppose
$\lambda_0,\lambda_1\colon|\bar\pi|\to\calK(\bigcirc)$ are two ways of
lifting plane curves into knots in $\PDS_p$ (namely, $P\circ\lambda_i=I$).
Then for $\gamma\in|\bar\pi|$,\hfill\text{\scriptsize {\red\bf Lemma 1.} ``Division by $\hbar$'' is
well-defined.}
\[ \eta(\gamma) \coloneqq
  (\lambda_0(\gamma)-\lambda_1(\gamma))/\hbar\in\calK^{/1}_H(\bigcirc\bigcirc)
  = |\bar\pi|\otimes|\bar\pi|
\]
and we get an operation $\eta$ on plane curves. If Kontsevich likes
$\lambda_0$ and $\lambda_1$ (namely if there are $\lambda^a_i$ with
$Z^{/2}(\lambda_i(\gamma)) = \lambda^a_i(W(\gamma))$), then $\eta$ will have a compatible algebraic
companion $\eta^a$:
\[ \eta^a(\alpha) \coloneqq
  (\lambda^a_0(\alpha)-\lambda^a_1(\alpha))/\hbar\in\calA^{/1}_H(\bigcirc\bigcirc)
  = |A|\otimes|A|.
\]
For indeed, in $\calA_H^{/2}$ we have
$
  \hbar W(\eta(\gamma))
  = \hbar Z(\eta(\gamma))
  = Z(\lambda_0(\gamma)) - Z(\lambda_1(\gamma))
  = \lambda^a_0(W(\gamma)) - \lambda^a_1(W(\gamma))
  = \hbar\eta^a(W(\gamma)).
$
}}}}

\def\ExampleA{{\raisebox{2mm}{\parbox[t]{3.45in}{
\parshape 4 0in 1.65in 0in 1.65in 0in 1.65in 0in \linewidth
{\red\bf Example 1.} With $\gamma_1,\gamma_2\in|\pi|$ (or $|\bar\pi|$)
set $\lambda_0(\gamma_1,\gamma_2)=\tilde\gamma_1\cdot\tilde\gamma_2$
and $\lambda_1(\gamma_1,\gamma_2)=\tilde\gamma_2\cdot\tilde\gamma_1$
where $\tilde\gamma_i$ are arbitrary lifts of $\gamma_i$. Then $\eta_1$
is the Goldman bracket! Note that here $\lambda_0$ and $\lambda_1$
are not well-defined, yet $\eta_1$ is.
}}}}

\def\ExampleB{{\raisebox{2mm}{\parbox[t]{2.95in}{
{\red\bf Example 2.} With $\gamma_1,\gamma_2\in\pi$ (or $\bar\pi$) and with $\lambda_0,\lambda_1$ as on the right, we get the ``double
bracket'' $\eta_2\colon\pi\otimes\pi\to\pi\otimes\pi$ (or $\bar\pi\otimes\bar\pi\to\bar\pi\otimes\bar\pi$).
}}}}

\def\ExampleC{{\raisebox{2mm}{\parbox[t]{2.95in}{
\parshape 4 0in 1.9in 0in 1.9in 0in 1.9in 0in \linewidth
{\red\bf Example 3.} With $\gamma\in\bar\pi$ and $\lambda_0(\gamma)$
its ascending realization as a bottom tangle and $\lambda_1(\gamma)$
its descending realization as a bottom tangle, we get
$\eta_3\colon\bar\pi\to\bar\pi\otimes|\bar\pi|$. Closing the first
component and anti-symmetrizing, this is the Turaev cobracket.
}}}}

\def\ExampleD{{\raisebox{2mm}{\parbox[t]{2.7in}{
{\red\bf Example 4} \cite{Massuyeau:FormalDescription}. With $\gamma\in\bar\pi$ and $\lambda_0(\gamma)$
its ascending outer double and $\lambda_1(\gamma)$ its ascending inner
double we get $\eta_4\colon\bar\pi\to\bar\pi\otimes\bar\pi$. After some
massaging, it too becomes the Turaev cobracket.
}}}}

\def\TheRest{{\raisebox{-0.5mm}{\parbox[t]{3.95in}{
The rest is essentially {\bf\red Exercises:} {\red\bf 1.}~Lemma~1?
\quad {\red\bf 2.}~$\calA$?
\quad {\red\bf 3.}~\text{Fact~2?}
\quad {\red\bf 4.}~$\calA^{/1}$? Especially, $\calA^{/1}(\bigcirc)\cong|A|$!
\quad {\red\bf 5.}~Explain why Kontsevich likes our $\lambda$'s.
\quad {\red\bf 6.}~Figure out $\eta^a_i$, $i=1,\ldots,4$.
}}}}

\def\Kontsevich{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Kontsevich in a Pole Dance Studio.}\hfill(w/o poles? See~\cite{Kontsevich:Vassiliev,
Bar-Natan:OnVassiliev})

\parshape 1 0in 3.2in \vskip -4mm
\[ Z = \left(
    \sum_{m=0}^\infty \frac{1}{(2\pi i)^m}
    \hspace{-3mm}
    \sumint_{\stackrel{t_1<\ldots<t_m}{P=\{(z_i,z'_i)\}}}
    \hspace{-3mm}
    (-1)^{\#P_{\downarrow}}D_P
    \bigwedge_{i=1}^{m}\frac{dz_i-dz'_i}{z_i-z'_i}
  \right)^\sim\in\calA
\]
}}}}

\def\KonCom{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Comments on the Kontsevich Integral.}
\begin{enumerate}
\item In the tangle case, the endpoints are fixed at top and bottom.
\item The $(\cdots)^\sim$ means ``a correction is needed near the caps and the cups'' (for the framed
  version, see~\cite{LeMurakami:Framed, Dancso:Kontsevich}).
\item There are never $pp$ chords, and no $4T_{pps}$ and $4T_{ppp}$ relations.
\item $Z$ is an ``expansion''.
\item $Z$ respects the $ss$ filtration and so descends to $Z^{/s}\colon\calK^{/s}\to\calA^{/s}$.
\end{enumerate}
}}}}

\def\ACom{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Comments on $\calA$.} In $\calA^{/1}$ legs on poles commute,
\newline so $\calA^{/1}(\bigcirc)=|A|$!
\vskip 1.5mm
In $\calA_H^{/2}$ we have:
}}}}

\def\ExampleAa{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Example $1^a$.} $\eta_1^a(|xyxy|,|xyx|)=$
}}}}

\def\ExampleCa{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Example $3^a$.} Ignoring complications, $\eta_3^a(xxyxyx)=$
}}}}

\def\Unignoring{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Unignoring the Complications.} We need $\lambda_0$ and $\lambda_1$ such that:
\begin{enumerate}
\item $\lambda_1(\gamma)$ is obtained from $\lambda_0(\gamma)$ by flipping all self-intersections from
ascending to descending.
\item Up to conjugation, $\lambda_1(\gamma)$ is obtained from $\lambda_0(\gamma)$ by a global flip.
\item $Z(\lambda_i(\gamma))$ is computable from $W(\gamma)$ and $Z^{/1}(\lambda_i(\gamma))=W(\gamma)$.
\end{enumerate}
}}}}

\def\lamA{{$\lambda_0(\gamma)=$}}
\def\lamB{{$\lambda_1(\gamma)=$}}

\def\ProofOfLemmaA{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\red\bf Proof of Lemma 1.} We partially prove Theorem~2 instead:

{\red\bf Theorem 2.} $\operatorname{gr}^\bullet\calK_H \cong
\bbF\llbracket\hbar\rrbracket\otimes(\calK^{/1})_0$.

{\red\bf Proof mod $\hbar^2$.} The map $\leftarrow$ is obvious. To go $\rightarrow$, map
$\calK_H\to\bbF\llbracket\hbar\rrbracket\otimes\calK^{/1}$ using
$\overcrossing\mapsto\virtualcrossing+\frac{\hbar}{2}\upupsmoothing$ and
$\undercrossing\mapsto\virtualcrossing-\frac{\hbar}{2}\upupsmoothing$ and apply
the functor $\operatorname{gr}^\bullet$.
}}}}

\def\Homework{{\raisebox{2mm}{\parbox[t]{3.95in}{
\begin{enumerate}[left=5pt .. 16pt]
\item Is there more than Examples 1--4?
\item Derive the bialgebra axioms from this perspective.
\item What more do we get if we don't mod out by HOMFLY-PT?
\item What more do we get if we allow more than one strand-strand interaction?
\item In this language, recover Kashiwara-
  \newline Vergne \cite{AKKN1, AKKN2}.
\item How is all this related to w-knots?
\item Do the same with associators. Use that to derive formulas for solutions of Kashiwara-Vergne.
\item What's the relationship with the Habiro-Massuyeau invariants of
  links in handlebodies \cite{HabiroMassuyeau:BottomTangles} (different filtration!).
\item Pole dance on other surfaces!
\item Explore the action of the mapping class group.
\end{enumerate}
}}}}

\def\Acknowledgement{{\raisebox{2mm}{\parbox[t]{3.95in}{\small
{\red\bf Acknowledgement.} This work was partially supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).
}}}}

\def\refs{{\raisebox{5mm}{\parbox[t]{3.95in}{
%{\red\bf References.}
{\footnotesize
%\par\vspace{-3mm}
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\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}
\def\nl{{\newline}} \import{../LesDiablerets-2208}{refs.tex}
\end{thebibliography}}
}}}}

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