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\begin{document} %\latintext
\newdimen\captionwidth\captionwidth=\hsize
\setcounter{secnumdepth}{4}

\title{A Perturbed-Alexander Invariant}

\author{Dror~Bar-Natan}
\address{
  Department of Mathematics\\
  University of Toronto\\
  Toronto Ontario M5S 2E4\\
  Canada \ifpub{(corresponding author)}{}
}
\email{drorbn@math.toronto.edu}
\urladdr{http://www.math.toronto.edu/drorbn}

\author{Roland~van~der~Veen}
\address{
  University of Groningen, Bernoulli Institute\\
  P.O. Box 407\\
  9700 AK Groningen\\
  The Netherlands \ifpub{(corresponding author)}{}
}
\email{roland.mathematics@gmail.com}
\urladdr{http://www.rolandvdv.nl/}

\date{First edition ?, this edition \today.}

\subjclass[2010]{TBW}
\keywords{TBW}

\thanks{This work was partially supported by NSERC grant RGPIN-2018-04350. It is available in electronic form, along with source files and a verification {\sl Mathematica} notebook at \url{http://drorbn.net/APAI} and at \arXiv{2201.?????}.}

\begin{abstract}
In this research announcement we give concise formulas, which lead to a simple and fast computer program, which computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot $K$ we write one of the standard matrices $A$ whose determinant is the Alexander polynomial of $K$, yet instead of computing its determinant we consider a certain quadratic expression in the entries of $A^{-1}$. The proximity of our formulas to the Alexander polynomial suggest that they should have a topological explanation. This we don't have yet.
\end{abstract}

\maketitle

%\setcounter{tocdepth}{3}
%\tableofcontents
%\eject


\section{The Formulas} The selling point for this article is that the formulas in it are concise. Thus we start by running through these formulas for a knot invariant $\rho_1$ as quickly as we can. In Section~\ref{sec:Implementation} we turn the formulas into a short yet very fast computer program, and in Section~\ref{sec:Context} we quickly sketch the context: Alexander, Burau, Jones, Melvin, Morton, Rozansky, Overbay, Ohtsuki, and our own prior work.

\parpic[r]{\input{SampleKnot.pdf_t}}
Given an oriented $n$-crossing knot $K$, we draw it in the plane as a long knot diagram $D$ in such a way that the two strands intersecting at each crossing are pointed up (that's always possible because we can always rotate crossings as needed), and so that at its beginning and at it's end the knot is oriented upward. An example is here on the right. We then label each edge of the resulting diagram with two integer labels: a running index $k$ which runs from 1 to $2n+1$, and a ``rotation number'' $r_k$, which counts how many times the geometric rotation number of that edge (the signed number of times the tangent to the edge is horizontal and heading right, with cups counted with $+1$ signs and cups with $-1$; this number is well defined because at their ends, all edges are headed up). On the right the running index runs from $1$ to $7$, and the rotation numbers for all edges is $0$ except for $r_4$, for which it is $-1$.

Next, we form a $(2n)\times(2n+1)$ matrix $B$ of Laurent polynomials in a formal variable $T$ as follows. Every crossing $c$ of $K$ has a sign $s\in\{\pm 1\}$ and is surrounded by four edges, with labels $i$ and $j$ below the crossing (where the label $i$ belongs to the over strand and $j$ to the under strand) and labels $k=i+1$ and $l=j+1$ above the crossing. Such a crossing defines rows $i$ and $j$ and columns $i$, $j$, $k$ and $l$ of $B$ as below, with the rest of $B$ set to be $0$:
\begin{equation} \label{eq:B}
  \begin{array}{c}\input{Xings.pdf_t}\end{array}
  \qquad\longrightarrow\qquad
  \begin{array}{c|cccc}
    B &   \text{col }i & \text{col }j &  \text{col }k  &  \text{col }l \\
    \hline
    \text{row }i & 1 & 0 & -1  & 0 \\
    \text{row }j & 0 & 1& T^s-1  & -T^s
  \end{array}
\end{equation}

\parpic[r]{$\ds
  G=\left(\begin{array}{ccc}
    0 & \cdots & 0 \\
    &&\\
    & A^{-1} & \\
    &&
  \end{array}\right)
$}
We then remove the first column of $B$ and call the result $A$ (so $B=\left(\phi|A\right)$, with $\phi$ a single column), we invert $A$ and prepend to it a row of $0$'s, and we call the resulting $(2n+1)\times(2n)$ matrix $G=(g_{\alpha\beta})$, as on the right. In the case of the trefoil knot as shown above, the matrices $B$ and $G$ are
\[ \tiny % Matrices produced by the notebook APAI.nb
\left(
\begin{array}{ccccccc}
 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
 0 & 1 & -T & 0 & 0 & T-1 & 0 \\
 0 & 0 & 1 & -1 & 0 & 0 & 0 \\
 0 & T-1 & 0 & 1 & -T & 0 & 0 \\
 0 & 0 & 0 & 0 & 1 & -1 & 0 \\
 0 & 0 & 0 & T-1 & 0 & 1 & -T \\
\end{array}
\right),
\ \left(
\begin{array}{cccccc}
 0 & 0 & 0 & 0 & 0 & 0 \\
 -1 & 0 & 0 & 0 & 0 & 0 \\
 -1 & -\frac{T}{T^2-T+1} & \frac{1-T}{T^2-T+1} & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & 0 \\
 -1 & -\frac{T}{T^2-T+1} & -\frac{T^2}{T^2-T+1} & \frac{1-T}{T^2-T+1} & -\frac{(T-1) T}{T^2-T+1} & 0 \\
 -1 & \frac{1}{-T^2+T-1} & -\frac{T}{T^2-T+1} & -\frac{T}{T^2-T+1} & \frac{1-T}{T^2-T+1} & 0 \\
 -1 & \frac{1}{-T^2+T-1} & -\frac{T}{T^2-T+1} & -\frac{T}{T^2-T+1} & -\frac{T^2}{T^2-T+1} & 0 \\
 -1 & -\frac{1}{T} & -1 & -\frac{1}{T} & -1 & -\frac{1}{T} \\
\end{array}
\right).
\]

\picskip{0}
We note without supplying details that the matrix $B$ comes in a straightforward way from Fox calculus as it is applied to the Wirtinger presentation of the fundamental group of the complement of $K$ (using the diagram $D$), and hence the determinant of $A$ is equal up to a unit to the Alexander polynomial of $K$. In fact, we have that
\begin{equation} \label{eq:Delta} \Delta = T^{(r(D)-w(D))/2}\det(A), \end{equation}
where $r(D)\coloneqq\sum_kr_k$ is the total rotation number of $D$, where $w(D)=\sum_cs_c$ is the writhe of $D$, namely the sum of the signs $s_c$ of all the crossings $c$ in $D$, and where $\Delta$ is the normalized Alexander polynomial of $K$, so $\Delta(T)=\Delta(T^{-1})$ and $\Delta(1)=1$. In our example $\det(A)=T^3-T^2+T$, $r(D)=-1$, and $w(D)=3$, so $\Delta=T^{(-1-3)/2}(T^3-T^2+T)=T-1+T^{-1}$, as expected for the trefoil knot.

We can now define our invariant $\rho_1$. It is the sum of two sums. The first is over all crossings $c$ in $D$, and for such a crossing we let $s$ denote its sign and we let $i$ and $j$ denote the edge labels of the incoming over- and under-strands, respectively. The second is a sum over the edges $k$ of $D$ of a correction term dependent on the rotation number $r_k$. We multiply both of these summands by $\Delta^2$ to ``clear the denominators''\footnote{The first summand is quadratic in the entries of $G$ and hence it has denominators proportional to $\Delta^2$.}:
\begin{equation} \label{eq:rho1} \begin{split} \rho_1 \coloneqq
  \Delta^2\sum_c s \left(\left(1-T^s\right)g_{ij}\left(g_{ij}-g_{jj}\right)+2g_{ii}g_{ij}-g_{ij}g_{ji}-g_{ii}g_{jj} - g_{ij} + g_{jj}  - 1/2\right) \\
  + \Delta^2\sum_kr_k\left(g_{kk}-1/2\right).
\end{split} \end{equation}
A direct calculation shows that in our example $\rho_1 = -T^2 +2T - 2 + 2/T - 1/T^2$.

\begin{theorem} \label{thm:Main} The quantity $\rho_1$ is a knot invariant.
\end{theorem}

This is a research announcement, so we will not prove Theorem~\ref{thm:Main} here. We merely comment that $\rho_1$ has more separation power than the Jones polynomial, yet it is closer to the more topologically-meaningful Alexander polynomial $\Delta$: it is cooked up from the same matrix $A$ and in terms of computational complexity, computing $\rho_1$ is not very different from computing $\Delta$. In order to compute $\Delta$ we need to compute the determinant of $A$, while to compute $\rho_1$ we need to invert $A$ and then compute a sum of $O(n)$ terms that are quadratic in the entries of $A^{-1}$. We have computed $\rho_1$ for knots with over 200 crossings using the unsophisticated implementation presented in Section~\ref{sec:Implementation}.

Topologists should be intrigued! $\rho_1$ is cooked from the same matrix as the Alexander polynomial $\Delta$, yet we have no topological interpretation for $\rho_1$.

\section{Implementation} \label{sec:Implementation}

\input APAINB.tex

MORE.

\section{Some Context} \label{sec:Context} TBW.

\vskip 2mm\noindent{\bf Acknowledgement.} We wish to thank TBW comments and suggestions.

\begin{thebibliography}{BDV}
\backrefparscanfalse\renewcommand*\backref[1]{See pp.~#1.}%
\renewcommand*\backrefalt[4]{\ifcase #1\or In page #2.\else In pages #2.\fi}

\bibitem[BDV]{Self} D.~Bar-Natan and R.~van der Veen,
  {\em A Perturbed-Alexander Invariant,}
  (self-reference), paper and related files at
  \url{http://drorbn.net/APAI}. The \arXiv{????.?????} edition may be older. \backrefprint

  \bibitem[BM]{Bar-NatanMorrison:KnotTheory} D.~Bar-Natan, S.~Morrison, and others,
  {\tt KnotThery`}, a knot theory mathematica package,
  \url{http://katlas.org/wiki/The_Mathematica_Package_KnotTheory}. \backrefprint

\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}. \backrefprint

\bibitem[Wo]{Wolfram:Mathematica}
  {\em Wolfram Language \& System Documentation Center,}
  \web{Wolf}. \backrefprint

\end{thebibliography}

\end{document}
\endinput 
