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    {\normalfont \bfseries Very Fast and Very Strong}
    
 %\medskip

%\parpic[r]{\scriptsize\normalfont \itshape February 18, 2025.}
    {\scriptsize Dror Bar-Natan} \\[-2pt]
    {\scriptsize University of Toronto, Department of Mathematics}%\\[-2pt]
   % {\scriptsize \normalfont \ttfamily drorbn@utoronto.ca}
    
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{\itshape February 19, 2025.} There is a knot invariant $\Theta$ that can go by a fancy name ``the two loop contribution to the Kontsevich integral'' [1--4].

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{\bfseries Theorem.} A down-to-earth algorithm for computing $\Theta$ exists that makes it computable for knots with hundreds of crossings [5].

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

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{\bfseries Strongly Supported Conjecture.} There is a {\itshape Seifert Formula}: $\Theta$ 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. In particular, $\Theta$ may say something about ribbon knots, whose Seifert surfaces (right) are pretty.

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 As a two variable polynomial,  $\Theta$ 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 obverse are the pictures corresponding to $\Theta$ for 15 random knots with 101--115 crossings. There are patterns there; we don't understand them yet.

 \scriptsize
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  \bibitem{Rozansky:U1RCC} 
  Rozansky.
  \newblock {A Universal $U(1)$-RCC Invariant of Links and Rationality Conjecture}.
  Preprint.         
           
\bibitem{GaroufalidisRozansky:LoopExpansion}
Garoufaldis  \&amp; Rozansky.
\newblock {The Loop Expansion of the Kontsevich Integral, the Null-Move, and $S$-Equivalence}.
Preprint.

\bibitem{Kricker:Lines} 
Kricker.\!
 \newblock  {The\! Lines\! of\! the\! Kontsevich\! Integral\! and\! Rozansky's\! Rationality\! Conjecture}.
  Preprint.

\bibitem{Ohtsuki:TwoLoop} 
Ohtsuki.
  \newblock{On the 2â€“loop Polynomial of Knots}.
  Geom. Top., 2007.

\bibitem{Theta} 
Bar-Natan \&amp; van~der~Veen.
  \newblock{A Very Fast, Very Strong, Topologically Meaningful and Fun Knot Invariant}.
  In preparation, \ttfamily{https://drorbn.net/Theta}.



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 \hfill {\scriptsize \, \textcopyright 2025 Mathematical Research Postcards}
  
  
    
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