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\def\myurl{http://www.math.toronto.edu/~drorbn}
\def\thistalk{Bonn-2505}
\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{{Happy birthday, dear Stavros!}}
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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 and far from rigorous.

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

\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 Alexander's polynomial and $\rho_1$ is Rozansky-Overbay's
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.
\hfill{\red Proof.} Use Fubini (details later).
}}}}

\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$\ds
  = \frac12\int_{\bbR^n}dy\,g_{ij}(\partial_{x_i}\!-\!\partial_{y_i})(\partial_{x_j}\!-\!\partial_{y_j})
    \exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right)$
\newline\null\hfill$\ds
  = \frac{1}{2\lambda^2}
    \int_{\bbR^n}\hspace{-2mm}dy\left(
      g_{ij}a^{ii'}\!a^{jj'}\!y_{i'}y_{j'} + \lambda g_{ij}a^{ji}
    \right)\exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right)$
\newline\null\hfill$\ds
  = \frac{1}{2\lambda^2}
    \int_{\bbR^n}dy\left(
      a^{ij}y_iy_j + \lambda n
    \right)\exp\left(-\frac{1}{2\lambda}a^{ij}y_iy_j + V(x+y)\right)$
\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 {\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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\newgeometry{textwidth=8in,textheight=10.5in}

\begin{multicols}{2}

\import{../Geneva-2408}{IType1.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{../Geneva-2408}{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 1mm\rule{\linewidth}{1pt}\vspace{0mm}

\needspace{0.75in}
\parpic[r]{$
  {\includegraphics[height=0.65in]{../../Projects/Gallery/Schaveling.jpg}
    \atop\text{\scriptsize Schaveling}}
$}
{\bf\red The $sl_3^{/\eps^2}$ Example} (continues
Schaveling~\cite{Schaveling:Thesis}). 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{../Geneva-2408}{Rank2.tex}

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

\import{../Geneva-2408}{theta.tex}

\needspace{5cm}
The torus knot\newline $T_{22/7}$:
\vskip -10mm
\[ \includegraphics[width=0.8\linewidth]{../Geneva-2408/PolyPlotT22-7.png} \]

\needspace{5cm}
Last, a random 250 crossing knot (knot from N.~Dunfield; more\newline at \web{DK}):
\vskip -7mm
\[ \includegraphics[width=0.8\linewidth]{../Beijing-2407/PolyPlot4DK250.png} \]

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

\parpic[r]{$
  {\includegraphics[height=0.7in]{../../Projects/Gallery/Ohtsuki.jpg}
    \atop\text{\scriptsize Ohtsuki}}
  {\includegraphics[height=0.7in]{../../Projects/Gallery/Garoufalidis.jpg}
    \atop\text{\scriptsize Garoufalidis}}
  {\includegraphics[height=0.7in]{../../Projects/Gallery/Kashaev.jpg}
    \atop\text{\scriptsize Kashaev}}
$}
{\bf\red Prior Art.} $\theta$ is probably equal to the ``2-loop
polynomial'' studied by Ohtsuki at \cite{Ohtsuki:2Loop} (at much greater
difficulty, and with harder computations).  $\theta$ is related, but
probably not equivalent, to the invariant studied by Garoufalidis and
Kashaev at \cite{GaroufalidisKashaev:BraidedHopf}.

{\red\bf $\theta$ Sees Topology!} Indeed, for a knot $K$, half the $T_1$
degree (say) of $\theta(K)$ bounds the genus of $K$ from below, and this bound is sometimes
better (and sometimes worse) than the bound coming from $\Delta$. It is fair to hope that
``anything $\Delta$ can do $\theta$ can do too'' (see \cite{AKT}), and in particular, that
$\theta$ may say something about ribbon and/or slice properties.

\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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\parpic[r]{\includegraphics[width=0.4\linewidth]{../Groningen-240530/UniversalInvariant.jpg}}
{\bf\red The Philosophy Corner.} ``Universal invariants'', valued
in universal enveloping algebra (possibly quantized) rather than in
representations thereof, are a priori better than the representation
theoretic ones. They are compatible with strand doubling (the Hopf
coproduct), and as the knot genus and the ribbon property for knots are
expressible in terms of strand doubling, universal invariants stand a
chance to say something about these properties. Indeed, they sometimes
do! See e.g.\ \cite{AKT, Ohtsuki:2Loop, GaroufalidisKashaev:BraidedHopf,
LopezNeumannVanDerVeen:Genus, BecerraGarrido:Thesis}. Representation
theoretic invariants don't do that!

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{\footnotesize

\bibitem[BN1]{DPG} D.~Bar-Natan,
  {\em Everything around $sl_{2+}^\epsilon$ is DoPeGDO.}
  \hfill{\normalsize\red\bf References.}
  \newline{\em So what?,}
  talk given in ``Quantum Topology and Hyperbolic Geometry Conference'',
  Da Nang, Vietnam, May 2019. Handout and video at \web{DPG}.

\bibitem[BN2]{AKT} D.~Bar-Natan,
  {\em Algebraic Knot Theory,}
  talk given in Sydney, September 2019. Handout and video at \web{AKT}.

\bibitem[BN3]{Cars} D.~Bar-Natan,
  {\em Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants,}
  talk given in ``Using Quantum Invariants to do Interesting Topology'',
  Oaxaca, Mexico, October 2022. Handout and video at \web{Cars}.

\bibitem[BV1]{PP1} D.~Bar-Natan and R.~van~der~Veen,
  {\em A Polynomial Time Knot Polynomial,}
  Proc.\ Amer.\ Math.\ Soc.\ {\bf 147} (2019) 377--397, \arXiv{1708.04853}.

\bibitem[BV2]{APAI} D.~Bar-Natan and R.~van~der~Veen,
  {\em A Perturbed-Alexander Invariant,}
  to appear in Quantum Topology, \web{APAI}.

\bibitem[BV3]{PG} D.~Bar-Natan and R.~van~der~Veen,
  {\em Perturbed Gaussian Generating Functions for Universal Knot Invariants,}
  \arXiv{2109.02057}.

\bibitem[BG]{BecerraGarrido:Thesis} J.~Becerra~Garrido,
  {\em Universal Quantum Knot Invariants,}
  Ph.D.\ thesis, University of Groningen, \web{BG}.

\bibitem[GK]{GaroufalidisKashaev:BraidedHopf} S.~Garoufalidis and R.~Kashaev,
  {\em Multivariable Knot Polynomials from Braided Hopf Algebras with Automorphisms,}
  \arXiv{2311.11528}.

\bibitem[GST]{GompfScharlemannThompson:Counterexample} R.~E.~Gompf,
  M.~Scharlemann, and A.~Thompson,
  {\em Fibered Knots and Potential Counterexamples to the Property 2R and
    Slice-Ribbon Conjectures,}
  Geom.\ and Top.\ {\bf 14} (2010) 2305--2347, \arXiv{1103.1601}.

\bibitem[GPV]{GPV} M.~Goussarov, M.~Polyak, and O.~Viro,
  {\em Finite type invariants of classical and virtual knots,}
  Topology {\bf 39} (2000) 1045--1068, \arXiv{math.GT/9810073}.

\bibitem[La]{Lawrence:UniversalUsingQG} R.~J.~Lawrence,
  {\em Universal Link Invariants using Quantum Groups,}
  Proc.\ XVII Int.\ Conf.\ on Diff.\ Geom.\ Methods in Theor.\ Phys., Chester,
  England, August 1988. World Scientific (1989) 55--63.

\bibitem[LV]{LopezNeumannVanDerVeen:Genus} D.~L\'opez Neumann and R.~van~der~Veen,
  {\em Genus Bounds from Unrolled Quantum Groups at Roots of Unity,}
  \arXiv{2312.02070}.

\bibitem[Oh1]{Ohtsuki:QuantumInvariants} T.~Ohtsuki,
  {\em Quantum Invariants,}
  Series on Knots and Everything {\bf 29}, World Scientific 2002.

\bibitem[Oh2]{Ohtsuki:2Loop} T.~Ohtsuki,
  {\em On the 2–Loop Polynomial of Knots,}
  Geom.\ Topol.\ {\bf 11-3} (2007)  1357--1475.

\bibitem[Ov]{Overbay:Thesis} A.~Overbay,
  {\em Perturbative Expansion of the Colored Jones Polynomial,}
  Ph.D.\ thesis, University of North Carolina, August 2013, \web{Ov}.

\bibitem[R1]{Ro1} L.~Rozansky,
  {\em A Contribution of the Trivial Flat Connection to the Jones
  Polynomial and Witten's Invariant of 3D Manifolds, I,}
  Comm.\ Math.\ Phys.\ {\bf 175-2} (1996) 275--296, \arXiv{hep-th/9401061}.

\bibitem[R2]{Ro2} L.~Rozansky,
  {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton
    Expansion of the Colored Jones Polynomial,}
  Adv.\ Math.\ {\bf 134-1} (1998) 1--31, \arXiv{q-alg/9604005}.

\bibitem[R3]{Ro3} L.~Rozansky,
  {\em A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture,}
  \arXiv{math/0201139}.

\bibitem[Sch]{Schaveling:Thesis} S.~Schaveling,
  {\em Expansions of Quantum Group Invariants,}
  Ph.D.\ thesis, Universiteit Leiden, September 2020, \web{Scha}.

}

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