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    {\normalfont \bfseries A Very Fast and Very Strong Knot Invariant}
    
 \medskip

\parpic[r]{\scriptsize\normalfont \itshape February 18, 2025.}
    {\scriptsize Dror Bar-Natan} \\[-2pt]
    {\scriptsize University of Toronto}\\[-2pt]
    {\scriptsize \normalfont \ttfamily drorbn@utoronto.ca}
    
    \medskip
 
There is a knot invariant $\Theta$ that can go by a fancy name ``the two loop contribution to the Kontsevich integral'' [1--4].

\vskip -1mm
{\bfseries Theorem} [6]. There is a down to earth algorithm to compute $\Theta$ which makes it computable for knots with hundreds of crossings.

\vskip -1mm
{\bfseries Fact.} $\Theta$ is much stronger than the HOMFLY-PT polynomial $H$ and Khovanov homology $KH$ taken together. On the 313,230 prime knots with up to 15 crossings it attains 306,472 distinct values -- a deficit of 6,758 -- whereas $H$ and $KH$ together have a deficit of 70,245, about 10 times the worse.

\vskip -1mm
{\bfseries Strongly Supported Conjecture.} $\Theta(K)$ has a ``Seifert Formula'': It can be presented as a perturbed Gaussian integral of a Lagrangian on (6 copies of) the first homology $H_1$ of a Seifert surface $\Sigma$ of a knot $K$, itself defined using low degree finite type invariants of links representing  classes in $H_1$. Thus $\Theta$ bounds the genus of $K$.

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\parpic[r]{\includegraphics[width=0.75in]{Seifert4Ribbon.pdf}}
{\bfseries Dream.} Pretty Seifert surfaces will lead to pretty formulas, and in particular, $\Theta$ may say something about ribbon knots, whose Seifert surfaces, as on the right, are pretty.

\vskip -1mm
$\Theta$ is a two variable polynomial. A two variable polynomial is a 2D array of coefficients, which can be interpreted as directing the colours of a 2D array of pixels, which can be viewed as a picture. On the other side are the 15 pictures corresponding to $\Theta$ of 15 random knots with 101--115 crossings. There are patterns there; we don't understand them yet.

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%\def\bysame{{---}}
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\begin{thebibliography}{}

%\bibitem{Ro} 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.\ {\bfseries 175-2} (1996) 275--296, \arXiv{hep-th/9401061}.
%
%\bibitem{Rozansky:Burau} \bysame,
%  {\em The Universal $R$-Matrix, Burau Representation and the Melvin-Morton
%    Expansion of the Colored Jones Polynomial,}
%  Adv.\ Math.\ {\bfseries 134-1} (1998) 1--31, \arXiv{q-alg/9604005}.

\bibitem{Rozansky:U1RCC} L.~Rozansky,
  {\em A Universal $U(1)$-RCC Invariant of Links and \newline
  Rationality Conjecture,}
  \arXiv{math/0201139}.

\bibitem{GaroufalidisRozansky:LoopExpansion} S.~Garoufaldis, L.~Rozansky,
  {\em The Loop Expansion of the Kontsevich\newline
  Integral, the Null-Move, and $S$-Equivalence,}
  \arXiv{math.GT/0003187}.

\bibitem{Kricker:Lines} A.~Kricker,
  {\em The Lines of the Kontsevich Integral and Rozansky's\newline
  Rationality Conjecture,}
  \arXiv{math/0005284}.

\bibitem{Ohtsuki:TwoLoop} T.~Ohtsuki,
  {\em On the 2–loop Polynomial of Knots,}
  Geom.\ Top.\ {\bfseries 11}\newline (2007) 1357--1475.

\bibitem{Theta} DBN, R.~van~der~Veen,
  {\em A Very Fast, Very Strong, Topologically\newline
  Meaningful and Fun Knot Invariant,}
  in preparation at \url{https://drorbn.net/Theta}.

\end{thebibliography}

\vfill
 \hfill {\scriptsize \, \textcopyright 2024 Mathematical Research Postcards}
    
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