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\setlist[itemize]{left=0pt .. 10pt}
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% 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{ICERM-2305}
\def\title{{\purple Shifted} Partial Quadratics, their Pushforwards, and Signature Invariants for Tangles}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to ICERM!}}
\def\zoom{{Zoomers: Please please pretty please, webcams on!}}
\def\webdef{{\href{http://drorbn.net/icerm23}{http://drorbn.net/icerm23}}}
\def\web#1{{\href{\myurl/Talks/\thistalk/#1}{{\greektext web}/#1}}}
\def\titleA{{\title}}
\def\titleB{{\title}}
\def\titleC{{\title}}

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\def\pink{\color{pink}}
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\def\purple{\color{purple}}
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\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\barF{{\bar F}}
\def\bart{{\bar t}}
\def\barT{{\bar T}}
\def\barX{{\bar X}}
\def\bbC{{\mathbb C}}
\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\calC{{\mathcal C}}
\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}}

\newcommand{\pluseq}{\mathrel{{+}{=}}}
\newcommand{\minuseq}{\mathrel{{-}{=}}}

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

%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.33in}{
{\red\bf Abstract.} Following a general discussion of the computation of
zombians of unfinished columbaria (with examples), I will tell you about
my recent joint work w/ Jessica Liu on what we feel is the ``textbook''
extension of knot signatures to tangles, which for \text{unknown} reasons,
is not in any of the textbooks that we know.
}}}}

\def\freepik{{Image: \href{http://freepik.com}{Freepik.com}}}

\def\PriorArt{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Prior Art} on signatures for tangles / braids.
Gambaudo and Ghys~\cite{GambaudoGhys:BraidsAndSignatures},
Cimasoni and Conway~\cite{CimasoniConway:ColoredTanglesAndSignatures},
Conway~\cite{Conway:SignaturesSurvey},
Merz~\cite{Merz:CimasoniConway}. All define signatures of tangles / braids by first closing them to
links and then work hard to derive composition properties.
}}}}

\def\WhyTangles{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Why Tangles?} $\bullet$ Faster!
\newline$\bullet$ Conceptually clearer proofs of invariance
\newline\null\quad (and of skein relations).
\newline$\bullet$ Often fun and consequential:
\newline\null\ $\circ$ The Jones Polynomial $\leadsto$ The Temperley-Lieb Algebra.
\newline\null\ $\circ$ Khovanov Homology $\leadsto$ ``Unfinished complexes'', complexes
\newline\null\qquad in a category.
\newline\null\ $\circ$ The Kontsevich Integral
\newline\null\qquad $\leadsto$ Associators.
\newline\null\ $\circ$ HFK $\leadsto$ OMG, type $D$,
\newline\null\qquad type $A$, $\calA_\infty$, \ldots
}}}}

\def\savings{{$2^{n/2}+2^{n/2}+2^{\sqrt{n}} \ll 2^n$}}

\def\Computing{{\raisebox{2mm}{\parbox[t]{2.87in}{
\parshape 1 0in 4in
{\red\bf Computing Zombians of Unfinished Columbaria.}
\begin{itemize}
\item Must be no slower than for finished ones.
\item Future zombies must be able to complete the computation.
\item Future zombies must not even know the size of the task that today's zombies were facing.
\item We must be able to extend to ZPUCs, Zombie Processed Unfinished Columbaria!
\end{itemize}

\parshape 1 0in 2.87in
{\red\bf Example / Exercise.} Compute the determinant of a $1,000\times 1,000$ matrix in which $50$
entries are not yet given.

\parshape 1 0in 3.96in
{\red\bf Homework / Research Projects.} $\bullet$ What with
ZPUCs? $\bullet$ Use this to get an Alexander tangle invariant.
}}}}

\def\zfreepik{{Zombies: \href{http://freepik.com}{Freepik.com}}}

\def\Reminders{{\raisebox{4mm}{\parbox[t]{3.96in}{
{\red\bf Reminders.} $\{\text{knots}\} \rightrightarrows \{\text{matrices / quadratic
forms}\} \xrightarrow[\sigma]{\text{signature}} \bbZ$:
}}}}

\def\KasConj{{\raisebox{1.5mm}{\parbox[t]{0.69in}{
{\bf\red Kashaev's Conj.} \cite{Kashaev:SymmetricMatrices}
\newline\null\hfill$\sigma_{\text{\it Kas}}=2\sigma_{\text{\it TL}}$.
}}}}

\def\sss{$\sigma_+ \!-\! \sigma_- \!+\! s$}

\def\X{$X_{-i,j,k,-l}$}
\def\Xb{$\barX_{-i,j,k,-l}$}

\def\TLp{$A\pluseq
  \begin{pmatrix}
	-r	&	-t	& 2t	& \bart	\\
	-\bart	&	0	& \bart	& 0	\\
	2\bart	&	t	& -r	& -\bart\\
	t	&	0	& -t	& 0
  \end{pmatrix}
  \begin{matrix}i\\j\\k\\l\end{matrix}
$}
\def\TLm{$A\pluseq
  \begin{pmatrix}
	r	&	-t	& -2\bart& \bart\\
	-\bart	&	0	& \bart	& 0	\\
	-2t	&	t	& r	& -\bart\\
	t	&	0	& -t	& 0
  \end{pmatrix}
  \begin{matrix}i\\j\\k\\l\end{matrix}
$}

\def\Kasp{$A\pluseq
  \begin{pmatrix}
    v & u & 1 & u \\
    u & 1 & u & 1 \\
    1 & u & v & u \\
    u & 1 & u & 1
  \end{pmatrix}
  \begin{matrix}i\\j\\k\\l\end{matrix}
$}
\def\Kasm{$A\minuseq
  \begin{pmatrix}
    v & u & 1 & u \\
    u & 1 & u & 1 \\
    1 & u & v & u \\
    u & 1 & u & 1
  \end{pmatrix}
  \begin{matrix}i\\j\\k\\l\end{matrix}
$}

\def\sp{$s\minuseq 1$}
\def\sm{$s\pluseq 1$}

\def\wtrvu{{where $|\omega|=1$, $t=1-\omega$, $r=t+\bart$, $v=\operatorname{Re}(\omega)$, and
$u=\operatorname{Re}(\omega^{1/2})$.}}

\def\QuadraticForms{{\raisebox{0mm}{\parbox[t]{3.96in}{
A {\red quadratic form} on a v.s.\ $V$ over $\bbC$ is a quadratic $Q\colon
V\to\bbC$, or a sesquilinear Hermitian $\langle\cdot,\cdot\rangle$
on $V\times V$ (so
$\langle x,y\rangle = \overline{\langle y,x\rangle}$ and
$Q(y)=\langle y,y\rangle$), or given a basis $\eta_i$ of
$V^\ast$, a matrix $A=(a_{ij})$ with $A=\bar{A}^T$ and $Q =
\sum a_{ij}\bar\eta_i\eta_j$. The {\red signature} $\sigma$ of
$Q$ is $\sigma_+-\sigma_-$, where for some $P$, $\bar{P}^TAP =
\text{diag}(1,\stackrel{\sigma_+}{\cdots},1,-1,\stackrel{\sigma_-}{\cdots},-1,0,\ldots)$.
}}}}

\def\PQ{{\raisebox{0mm}{\parbox[t]{3.96in}{
A {\red\em Partial Quadratic (PQ)} on $V$ is a quadratic $Q$
defined only on a subspace $\calD_Q\subset V$. We add PQs with
$\calD_{Q_1+Q_2}\coloneqq\calD_{Q_1}\cap\calD_{Q_2}$. Given a linear $\psi\colon V\to W$ and a PQ $Q$
on $W$, there is an obvious {\em pullback} $\psi^*Q$, a PQ on $V$.

{\bf\red Theorem 1.} Given a linear $\phi\colon V\to W$ and a PQ $Q$ on $V$, there is a unique {\em
pushforward} PQ $\phi_*Q$ on $W$ such that for every $PQ$ $U$ on $W$,
$\yellowm{\sigma_V(Q+\phi^*U) = \sigma_{\ker\phi}(Q|_{\ker\phi}) + \sigma_W(U+\phi_*Q)}$.
\par {\small(If you must,
$\calD(\phi_*Q)=\phi(\operatorname{ann}_Q(\calD(Q)\cap\ker\phi))$
and $(\phi_*Q)(w)=Q(v)$, where $v$ is s.t.\ $\phi(v)=w$ and
$Q(v,\operatorname{rad} Q|_{\ker\phi})=0$).}
}}}}

\def\Gist{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Gist of the Proof.}
}}}}

\def\BT{$\bar{B}^T$}
\def\CT{$\bar{C}^T$}
\def\sigker{$\sigma(Q|_{\ker\phi})$}

\def\NowD{{\raisebox{-1mm}{\parbox[t]{3.96in}{
\ldots and the quadratic $F\eqqcolon\phi_*Q$ is well-defined only on $D\coloneqq\ker C$.
}}}}

\def\Exactly{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Exactly} what we want, if the Zombian is the signature!
\par $V$: The full space of {\em faces}.
\par $W$: The boundary, made of {\em gaps}.
\par $Q$: The known parts.
\par $U$: The part yet unknown.
\par $\sigma_V(Q+\phi^*(U))$: The overall Zombian.
\par $\sigma(Q|_{\ker\phi})$: An internal bit.  $U+\phi_*Q$: A boundary bit.
\par And so our ZPUC is the pair $S=(\sigma(Q|_{\ker\phi}), \phi_*Q)$.
}}}}

\def\SPQ{{\raisebox{0mm}{\parbox[t]{3.96in}{
A {\red\em Shifted Partial Quadratic (SPQ)} on $V$ is a pair $S=(s\in\bbZ, Q\text{ a PQ on }V)$. addition also adds the
shifts, pullbacks keep the shifts, yet $\phi_*S\coloneqq(s+\sigma_{\ker\phi}(Q|_{\ker\phi}),\phi_*Q)$ and
$\sigma(S)\coloneqq s+\sigma(Q)$.

{\bf\red Theorem 1' (\em Reciprocity).} Given $\phi\colon V\to W$, for SPQs $S$ on $V$ and $U$ on $W$ we have
$\yellowm{\sigma_V(S+\phi^*U) = \sigma_W(U+\phi_*S)}$ (and this characterizes $\phi_*S$).

\parshape 1 0in 3.333in
{\red\bf Theorem 2.} $\psi^*$ and $\phi_*$ are
functorial. Also, if $\alpha\act\beta = \gamma\act\delta$, $\alpha$ is surjective, $\beta$ is injective, and
$\operatorname{im}\gamma\supset\ker\delta$,
then $\gamma^*\act\alpha_* = \delta_*\act\beta^*$. Finally, $\psi^*$ is
additive but $\phi_*$ isn't.
}}}}

\def\pushpull{$\xymatrix@C=3mm@R=3mm{
  \bullet \ar[r]^\alpha \ar[d]_\gamma & \bullet \ar[d]^\beta \\
  \bullet \ar[r]_\delta \ar@{.>}[ru] & \bullet
}$}

\def\PlanarAlgebra{{\raisebox{2.5mm}{\parbox[t]{3.96in}{
{\red\bf Definition.} $\calS\begin{pmatrix}\input{figs/Tgaps.pdf_t}\end{pmatrix} \coloneqq
  \left\{\parbox{0.44in}{
    SPQ $S$ \\ on $\langle g_i\rangle$
  }\right\}$.

\parshape 6 0in 2.4in 0in 2.4in 0in 2.4in 0in 2.4in 0in 2.4in 0in 3.96in 
{\red\bf Theorem 3.} $\{\calS(\text{cyclic sets})\}$ is a planar
algebra, with compositions $\calS(D)((S_i))\coloneqq\phi^D_*(\psi_D^*(\bigoplus_i S_i))$,
where $\psi_D\colon\langle f_i\rangle\to\langle g_{\alpha i}\rangle$
maps every face of $D$ to the sum of the input gaps adjacent to it and
$\phi^D\colon\langle f_i\rangle\to\langle g_i\rangle$ maps every face
to the sum of the output gaps adjacent to it. So for our $D$,
$\psi_D$ is {\footnotesize
  $f_1\mapsto g_{34}$, $f_2\mapsto
  g_{31}+g_{14}+g_{24}+g_{33}$, $f_3\mapsto g_{32}$, $f_4\mapsto
  g_{11}$, $f_5\mapsto g_{13}+g_{21}$, $f_6\mapsto g_{23}$, $f_7\mapsto
  g_{12}+g_{22}$
} and $\phi^D$ is {\footnotesize
  $f_1\mapsto g_1$,
  $f_2\mapsto g_2+g_6$, $f_3\mapsto 0$, $f_4\mapsto g_3$, $f_5\mapsto 0$,
  $f_6\mapsto g_5$, $f_7\mapsto g_4$
}.

\parshape 1 0in 2.2in
{\red\bf Theorem 4.} {\it TL} and {\it Kas}, defined on $X$ and $\barX$ as before, extend to planar algebra morphisms
$\{\text{tangles}\}\to \{\calS\}$.
}}}}

\def\f#1{$f_{#1}$}
\def\g#1{$g_{#1}$}

% \def\Acknowledgement{{\raisebox{2mm}{\parbox[t]{3.95in}{
% }}}}

\def\refs{{\raisebox{5mm}{\parbox[t]{3.95in}{
%{\red\bf References.}
{\footnotesize
%\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}
\def\nl{} \input refs.tex
\end{thebibliography}}
}}}}

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

\begin{center}
\null\vfill\input{PQ1.pdftex_t}\vfill\null
\end{center}

\newgeometry{textwidth=8in,textheight=10.5in}

\def\Cordon{{
\includegraphics[width=0.2\linewidth]{figs/Cordon.pdf}
\hfill\includegraphics[width=0.75\linewidth]{Cordon.png}
\[
  \arraycolsep=0pt \renewcommand{\arraystretch}{1}
  \begin{array}{c} s \\ i \\ \null \end{array}
  \left(\begin{array}{c|cc}
    0 & \phi & C_{\text{rest}} \\ \hline
      \bar{\phi}^T & \lambda & \theta \\ \bar{C}_{\text{rest}}^T & \bar{\theta}^T & A_{\text{rest}}
  \end{array}\right)
  \to
  \arraycolsep=0pt \renewcommand{\arraystretch}{0}
  \begin{cases}
    \exists p\,\phi_p\neq 0 & \parbox{1.9in}{use $\phi_p$ to kill its row and \text{column}, drop a
      $\left(\begin{array}{cc}0&1\\[-5pt]1&0\end{array}\right)$ summand} \\
    \phi\!=\!0,\lambda\!\neq\!0 & \parbox{1.9in}{use $\lambda$ to kill $\theta$, let $s\pluseq\operatorname{sign}(\lambda)$} \\
    \phi\!=\!0,\lambda\!=\!0 & \parbox{1.9in}{append $\theta$ to $C_{\text{rest}}$.}
  \end{cases}
\]
}}

\begin{multicols*}{2}
\input{Signatures.tex}

\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}

\needspace{4mm} % 3mm is not enough
$\begin{CD}
  \begin{pmatrix} A & B \\ C & U \end{pmatrix}
  @>\det(A)>>
  \begin{pmatrix} I & A^{-1}B \\ C & U \end{pmatrix}
  @>1>>
  \begin{pmatrix} I & A^{-1}B \\ 0 & U-CA^{-1}B \end{pmatrix}
\end{CD}$,
\newline
  so $\det\begin{pmatrix} A & B \\ C & U \end{pmatrix} = \det(A)\det(U-CA^{-1}B)$.
  \hfill (what if $\not\exists A^{-1}$?)

\vskip 0mm\rule{\linewidth}{1pt}\vspace{-1mm}


%\setlist[enumerate]{left=0pt .. 12pt}

\needspace{20mm}
{\bf\red Questions.}
%\begin{enumerate}[left=5pt .. 17pt]
\begin{enumerate*}

\item Does this have a topological meaning?

\item Prove the Kashaev conjecture. Is there a version for tangles?

\item Find all solutions of R123 in our ``algebra''.

\item Braids and the Burau representation.

\item Recover the work in ``Prior Art''.

\item Are there any concordance properties?

\item What is the ``SPQ group''?

\item The jumping points of signatures are the roots of the Alexander polynomial. Does this
generalize to tangles?

\item Which of the three Cordon cases is the most common?

\item Are there interesting examples of tangles for which rels is non-trivial?

\item Is the $pq$ part determined by $\Gamma$-calculus?

\item Is the $pq$ part determined by finite type invariants?

\item Does it work with closed components / links?

\item Strand-doubling formulas?

\item A multivariable version?

\item Mutation invariance?

\item Ribbon knots?

\item Are there ``face-virtual knots''?

\item Does the pushforward story extend to ranks? To formal Gaussian measures? To super Gaussian measures?

\end{enumerate*}

\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}

\par{\red\bf References.}
\par\vspace{-3mm}
\renewcommand{\section}[2]{}%
\begin{thebibliography}{}
\setlength{\parskip}{0pt}
\setlength{\itemsep}{0pt plus 0.3ex}
\def\nl{}
\input refs.tex
\end{thebibliography}

\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}

{\bf\red Acknowledgement.} This work was partially supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).

\newpage

\input{Proofs.tex}

\end{multicols*}

\end{document}

\endinput

