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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Melbourne-2506}
\def\title{Knot Invariants from Zero-Dimensional QFT}

\def\navigator{{
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/Talks}{Talks}:
  \href{\myurl/Talks/\thistalk/}{\thistalk}:
}}
\def\thanks{{Thanks for inviting me to Melbourne!}}
\def\webdef{{\href{http://drorbn.net/mel25}{{\greektext web}$\coloneqq$http://drorbn.net/mel25}}}
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%%%

\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf Abstract.}
\parshape 1 0in 3.2in
For the purpose of today, an ``I-Type Knot Invariant'' is a knot invariant
computed from a knot diagram by integrating the exponential of a {\em
perturbed Gaussian} Lagrangian which is a sum over the features of that
diagram (crossings, edges, faces) of locally defined quantities, over
a product of finite dimensional spaces associated to those same features.

\parshape 1 0in 3.08in
{\bf Q.} Are there any such things?
\qquad{\bf A.} Yes.

\parshape 2 0in 3.96in 5mm 3.96in
{\bf Q.} Are they any good?
\quad{\bf A.} They are the strongest we know per CPU cycle, and are excellent in other ways too.

\hangindent=5mm\hangafter=2
{\bf Q.} Didn't Witten do that back in 1988 with path integrals?
\newline{\bf A.} No. His constructions are infinite dimensional\newline
  and far from rigorous.

{\bf Q.} But integrals belong in analysis!
\newline{\bf A.} Ours only use squeaky-clean algebra.
}}}}

\def\Dream{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf Dreams.}
\parshape 1 0in 2.95in
Given a knot $K$ with a Seifert surface $\Sigma$, we dream that
there is a 0D Lagrangian $L_\Sigma\colon 6H_1(\Sigma;\bbR)\to\bbR$
whose coefficients are (low degree) finite type invariants
of graphs representing multiple homology classes, such that
\[ Z=\underset{6H_1(\Sigma;\bbR)}{\Gint}\exp\left(L_\Sigma\right) \]

is the invariant $\theta$ presented below. We dream the formulas are
simple and natural, and that they have a universal generalization.

\parshape 1 0in 1.625in
{\red\bf Sweet. Even sweeter:} May say something about ribbon knots!
}}}}

\def\sltwoexample{{\bf\red The $sl_2^{/\eps^2}$ Example.} With $T$ an indeterminate and with
$\eps^2=0$:}

\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 =
  \underset{\bbR^{14}_{p_ix_i}\mathrlap{\ \text{
    measure on $\bbR$ is $(2\pi)^{-1/2}\cdot${\it standard}
  }}}{\Gint}
  {\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 $\calL(X^s_{ij}) = T^{s/2}\bbe^{L(X^s_{ij})}$ and
$\calL(C^\varphi_i) = T^{\varphi/2}\bbe^{L(C^\varphi_i)}$, and 
\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$\ds L(C^\varphi_i) = x_i(p_{i+1}-p_i) + \eps\varphi(1/2-x_ip_i)$
}}}}

\def\sltwoZ{{\raisebox{0mm}{\parbox[t]{3.96in}{
So $Z = T \Gint \bbe^{L(\righttrefoil)}dp_1\dots dp_7dx_1\dots dx_7$, where
$L(\righttrefoil)=$
$\ds
  \sum\nolimits_{i=1}^7 \hspace{-2mm} x_i(p_{i+1}-p_i)
  \ \ +\ \ (T-1)( {\red x_1(p_2-p_6)} + {\mgreen x_6(p_7-p_3)} + {\blue x_3(p_4-p_8)}) 
$
\null\hfill$\ds
  + \frac{\eps}{2}\left(\begin{array}{c}
    {\red x_1 (p_1-p_5) \left((T-1)x_1p_5+2(1-x_5p_5)\right)-1} \\
    + {\mgreen x_6 (p_6-p_2) \left((T-1)x_6p_2+2(1-x_2p_2)\right)-1} \\
    + {\blue x_3 (p_3-p_7) \left((T-1)x_3p_7+2(1-x_7p_7)\right)-1} \\
    + {\mpurple 2x_4p_4-1}
  \end{array}\right),
$
\newline and so $Z =
(T-1+T^{-1})^{-1}\exp\left(\eps\cdot\frac{(T-2+T^{-1})(T+T^{-1})}{(T-1+T^{-1})^2}\right)
=
\Delta^{-1}\exp\left(\eps\cdot\frac{(T-2+T^{-1})\rho_1}{\Delta^2}\right)$.
Here $\Delta$ is the Alexander polynomial and $\rho_1$ is the Rozansky-Overbay polynomial
\newline \cite{Ro1, Ro2, Ro3, Overbay:Thesis, PP1, APAI}.
}}}}

\def\Theorem{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\red\bf Theorem.} $Z$ is a knot invariant.
}}}}
\def\Proof{{\raisebox{2mm}{\parbox[t]{1.96in}{
{\red Proof.} Use Fubini.
}}}}

\def\AltGauss{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\bf\red (Alternative) Gaussian Integration.}

\parshape 1 0in 3.4in
{\bf Goal.} Compute
\hfill$\ds \int_{\bbR^n}dx \exp\left(-\frac12a^{ij}x_ix_j + V(x)\right).$\hfill\null

\vskip -3mm{\footnotesize (if convergent)}

{\bf Solution.} Set
\hfill$\ds
  \calZ_\lambda(x) \coloneqq
    \lambda^{-n/2}\int_{\bbR^n}dy \exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right).
$
Then $\calZ_1(0)$ is what we want, $\calZ_0(x) = (\det A)^{-1/2}\exp V(x)$, and
with $g_{ij}$ the inverse matrix of $a^{ij}$ and noting that under the $dy$ integral
$\partial_y=0$,
\hfill$\ds \frac12g_{ij}\partial_{x_i}\partial_{x_j}\calZ_\lambda(x) $
\newline\null\hfill$\ds
  = \frac{\lambda^{-n/2}}{2}\int_{\bbR^n}dy\,g_{ij}(\partial_{x_i}\!-\!\partial_{y_i})(\partial_{x_j}\!-\!\partial_{y_j})
    \exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right)$
\newline\null\hfill$\ds
  = \frac{\lambda^{-n/2}}{2\lambda^2}
    \hspace{-2mm}\int_{\bbR^n}\hspace{-2mm}dy\left(
      g_{ij}a^{ii'}\!a^{jj'}\!y_{i'}y_{j'} - \lambda g_{ij}a^{ji}
    \right)\exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right)$
\newline\null\hfill$\ds
  = \frac{\lambda^{-n/2}}{2\lambda^2}
    \int_{\bbR^n}dy\left(
      a^{ij}y_iy_j - \lambda n
    \right)\exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right)$
\newline\null\hfill$\ds
  = \partial_\lambda \calZ_\lambda(x).$
\newline Hence
\hfill (*)\quad$\ds \partial_\lambda \calZ_\lambda(x)
  = \frac12g_{ij}\partial_{x_i}\partial_{x_j}\calZ_\lambda(x),$
\newline and therefore
\hfill$\ds \calZ_\lambda(x) = (\det A)^{-1/2}
  \exp\left(\frac{\lambda}{2}g_{ij}\partial_{x_i}\partial_{x_j}\right) \exp V(x).
$
\vskip 3pt
\par We've just witnessed the birth of ``Feynman Diagrams''.

\parshape 3 0in 3.4in 0in 3.4in 0in 3.96in
{\bf\red Even better.} With $Z_\lambda\coloneqq\log(\sqrt{\det A}\calZ_\lambda)$, by a
simple substitution into (*), we get the ``Synthesis Equation'':
\vskip -4mm
\[ Z_0=V, \quad
  \partial_\lambda Z_\lambda = \frac12\sum\nolimits_{i,j=1}^ng_{ij}\left(
    \partial_{x_i,x_j}Z_\lambda + (\partial_{x_i}Z_\lambda)(\partial_{x_j}Z_\lambda)
  \right)
  \eqqcolon F(Z_\lambda),
\]
an ODE (in $\lambda$) whose solution is pure algebra.
}}}}

\def\PicardIteration{{\raisebox{2mm}{\parbox[t]{3.3in}{
{\bf\red Picard Iteration} (used to prove the existence and uniqueness of
solutions of ODEs). To solve $\partial_\lambda f_\lambda=F(f_\lambda)$
with a given $f_0$, start with $f_0$, iterate $f\mapsto f_0+\int_0^\lambda F(f_\lambda)d\lambda$,
and seek a fixed point. In our cases, it is always reached after finitely many iterations!
}}}}

\def\GintDef{{\raisebox{2mm}{\parbox[t]{3.96in}{
{\bf\red Definition.} $\Gint$: The result of this process, ignoring the convergence of the
actual integral.
}}}}

\def\Strong{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Strong.} The pair $(\Delta,\rho_1)$ attains
270,316 distinct values on the 313,230 prime knots with up to 15 crossings
(a deficit of 42,914), whereas the pair ($H=$ HOMFLYPT polynomial,
$Kh=$ Khovanov Homology) attains only 242,985 distinct values on the same
knots (a deficit of 70,245). The pair $(\Delta,\theta)$, discussed later, has a deficit of
only 6,758, and the triple $(\Delta,\theta,\rho_2)$, of only 6,341.

Yet better than $(H,Kh)$ and other Reshetikhin-Turaev-Witten invariants
and knot homologies, $\Delta$, $\rho_1$, $\rho_2$ and $\theta$ can be computed in {\red\bf
polynomial time} (and hence, even for very large knots).

So ugly as the formulas may be (and $\theta$'s formulas are uglier),
these invariants are possibly {\red\bf the best we have!}
}}}}

\def\Acknowledgement{{\raisebox{2mm}{\parbox[t]{3.95in}{
{\bf\red Acknowledgement.} This work was supported by NSERC
grants RGPIN-2018-04350 and RGPIN-2025-06718 and by the Chu Family Foundation (NYC).
}}}}

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\begin{document} \latintext
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\null\vfill\input{0D1.pdftex_t}\vfill\null
\end{center}

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

\begin{multicols}{2}

\import{../Bonn-2505}{IType.tex}

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

{\red\bf There's more!} To get $sl_2$ invariants mod $\eps^3$, add
the following to $L(X^+_{ij})$, $L(X^-_{ij})$, and $L(C^\varphi_i)$,
respectively (and see More.nb at \web{ap} for the verifications):

\import{../Bonn-2505}{More.tex}

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

{\red\bf Even more!}
$\bullet$ The $sl_2$ formulas mod $\eps^4$ are in the last page of the handout of~\cite{Cars}.

$\bullet$ Using~\cite{GPV} we can show that every finite type invariant is I-Type.

$\bullet$ Probably, $\langle$Reshetikhin-Turaev$\rangle \subset \langle$I-Type$\rangle$
efficiently.

$\bullet$ Possibly, $\langle$Rozansky Polynomials$\rangle \subset \langle$I-Type$\rangle$
efficiently.

$\bullet$ Knot signatures are I-Type, at least mod 8.

$\bullet$ We already have some work on $sl_3$, and it leads to the strongest
genuinely-computable knot invariant presently known.

%\vskip 0mm\rule{\linewidth}{1pt}\vspace{0mm}
\newcolumn

{\bf\red The $sl_3^{/\eps^2}$ Example} \cite{Theta}. Here we have two formal variables
$T_1$ and $T_2$, we set $T_3\coloneqq T_1T_2$, we integrate over
6 variables for each edge: $p_{1i}$, $p_{2i}$, $p_{3i}$, $x_{1i}$,
$x_{2i}$, and $x_{3i}$.

\import{../Bonn-2505}{Rank2.tex}

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

\import{../Bonn-2505}{theta.tex}

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

  ${\includegraphics[height=0.7in]{../../Projects/Gallery/Ohtsuki.jpg} \atop \text{\scriptsize Ohtsuki}}$
  \hfill ${\includegraphics[height=0.7in]{../../Projects/Gallery/Garoufalidis.jpg} \atop \text{\scriptsize Garoufalidis}}$
  \hfill ${\includegraphics[height=0.7in]{../../Projects/Gallery/Rozansky.jpg} \atop \text{\scriptsize Rozansky}}$
  \hfill ${\includegraphics[height=0.7in]{../../Projects/Gallery/Kricker.jpg} \atop \text{\scriptsize Kricker}}$
  \hfill ${\includegraphics[height=0.7in]{../../Projects/Gallery/Schaveling.jpg} \atop \text{\scriptsize Schaveling}}$
  \hfill ${\includegraphics[height=0.7in]{../../Projects/Gallery/Kashaev.jpg} \atop \text{\scriptsize Kashaev}}$

\vskip 1mm

{\bf\red Prior Art.} $\theta$ is probably equal to the ``2-loop
polynomial'' studied by Ohtsuki \cite{Ohtsuki:2Loop} (at greater
difficulty, with harder computations), continuing B-N, Garoufalidis,
Rozansky, Kricker, and Schaveling \cite{Bar-NatanGaroufalidis:MMR,
GaroufalidisRozansky:LoopExpansion, Ro1, Ro2, Ro3, Kricker:Lines, Schaveling:Thesis}.
$\theta$ is related, but probably not equivalent, to the invariant
studied by Garoufalidis--Kashaev \cite{GaroufalidisKashaev:BraidedHopf}.

\end{multicols}

\needspace{5cm}
{\bf\red The Rolfsen Table of Knots.}
\vskip 2mm

\includegraphics[width=\linewidth]{../Beijing-2407/g250.png}

\begin{multicols}{2}

\par{\red\bf Where is it coming from?} The most honest answer is ``we don't know'' (and {\red
that's good!}). The second
most, ``undetermined coefficients for an ansatz that made sense''. The ansatz comes from the
following principles / earlier work:

{\bf Morphisms have generating functions.} Indeed, there is an isomorphism
\[ \calG \colon \Hom(\bbQ[x_i],\bbQ[y_j]) \to \bbQ[y_j]\llbracket\xi_i\rrbracket, \]
and by PBW, many relevant spaces are polynomial rings, though only as vector spaces.

{\bf Composition is integration.} Indeed, if $f \in \Hom(\bbQ[x_i],\bbQ[y_j])$ and
$g\in\Hom(\bbQ[y_j],\bbQ[z_k])$, then
\[ \calG(g\circ f) = \int \bbe^{-y\cdot\eta}fg\,dy\,d\eta \]

{\bf Use universal invariants.} These take values in a universal
enveloping algebra (perhaps quantized), and thus they are expressible
as long compositions of generating functions. See~\cite{Lawrence:UniversalUsingQG,
Ohtsuki:QuantumInvariants}.

{\bf ``Solvable approximation'' $\leadsto$ perturbed
Gaussians.} Let $\frakg$ be a semisimple Lie algebra, let
$\frakh$ be its Cartan subalgebra, and let $\frakb^u$ and $\frakb^l$
be its upper and lower Borel subalgebras. Then $\frakb^u$ has
a bracket $\beta$, and as the dual of $\frakb^l$ it also has
a cobracket $\delta$, and in fact, $\frakg\oplus\frakh \equiv
\operatorname{Double}(\frakb^u,\beta,\delta)$. Let $\frakg^+_\eps
\coloneqq \operatorname{Double}(\frakb^u,\beta,\eps\delta)$ (mod
$\eps^{d+1}$ it is solvable for any $d$). Then by \cite{PG,DPG}
(in the case of $\frakg = sl_2$) all the interesting tensors of
$\calU(\frakg^+_\eps)$ (quantized or not) are perturbed Gaussian with
perturbation parameter $\eps$ with with understood bounds on the degrees
of the perturbations.

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{\footnotesize \import{../Bonn-2505}{refs.tex} }

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\hfill{\red\bf Thanks for bearing with me!}

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