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\begin{document}
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\title{A Poly-Time Knot Polynomial Via Solvable Approximation}

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

\author{Roland~van~der~Veen}
\address{
  Mathematisch Instituut\\
  Universiteit Leiden\\
  Niels Bohrweg 1\\
  2333 CA Leiden\\
  The Netherlands
}
\email{roland.mathematics@gmail.com}
\urladdr{http://www.rolandvdv.nl/}

\date{First edition Not Yet, 2017, this edition \today. Electronic version
and related files at~\cite{Self}, \url{\web}}

\subjclass[2010]{57M25}
\keywords{
  knots,
  tangles,
  knot polynomials,
  Lie algebras,
  Lie bialgebras}

\thanks{This work was partially supported by NSERC grant RGPIN 262178.}

\begin{abstract}
  \input abstract.tex
\end{abstract}

\maketitle

\setcounter{tocdepth}{3}
\tableofcontents

\section{Introduction}

An excellent classical paper in mathematics would state a great
theorem in the first couple pages of the introduction, describe its
significance in section 2, and then present its proof in the remaining
sections. We cannot match that; as a ``theorem'', our result is that a
certain algorithm works well. We cannot even state the theorem without describing
the algorithm, and a wordy description of the algorithm would either be
lengthy and unmotivated, or would take up a whole paper. So instead of
a wordy description we provide a complete implementation, right below
in Figure~\ref{fig:Splash}, along with one usage example. Once it is
established that the algorithm fits in less than a page of code, we move
on to explain why we think it is valuable, and more importantly, how it
arises from a traditional construction applied to a non-traditional Lie
algebra which enables a non-traditional computational technique.

\begin{figure}
\begin{multicols}{2}
\noindent{\bf A Demo Program for $z_1$} \vskip 2mm
\par\noindent\includegraphics[width=\linewidth]{Snips/Program.pdf}
\captionsetup{width=0.9\linewidth}
\caption{
  A demo program computing $z_1$, some initial data, and a sample
  run on the 0-framed trefoil knot.  The program is written in {\sl
  Mathematica}~\cite{Wolfram:Mathematica} and is available at~\cite{Self}.
}
\label{fig:Splash}
\columnbreak
\par\noindent{\bf Initial Data} \vskip 2mm
\par\noindent\includegraphics[width=\linewidth]{Snips/Data.pdf}
\par\noindent{\bf The Trefoil} \vskip 2mm
\par\noindent\includegraphics[width=\linewidth]{Snips/Run.pdf}
\end{multicols}
\end{figure}

\begin{theorem} The program in Figure~\ref{fig:Splash} computes $z_1$,
a polynomial invariant of knots, in time polynomial in the crossing number.
\end{theorem}

The invariant $z_1$ is in fact quite strong --- explicit computations
show that it separates more knots in the standard knot tables than the
Alexander polynomial, the Jones polynomial, the HOMFLY-PT polynomial,
and Khovanov homology (even when these are combined)! See the
table in Section~\ref{sec:z1}.

{\bf This is exciting news.} The main reason is self-evident ---
$z_1$ is the first poly-time computable polynomial invariants of
knots since the Alexander polynomial, discovered nearly 90 years
ago~\cite[1928]{Alexander:TopologicalInvariants}. A further reason is
explained in Section~\ref{sec:ribbon} --- it seems that $z_1$ has a better
chance than anything else we know to detect potential counterexamples
to the ribbon-slice conjecture. Finally, $z_1$ is founded on some new
Lie-theoretic techniques, it seems to be possible to generalize it in
several directions, and many questions are raised. See Section~\ref{sec:questions}.

Our $z_1$ can be considered as a perturbation of another invariant, $z_0$,
which is somewhat weaker yet in many ways it is more valuable. It is
computed by a shorter program (see Figure~\ref{fig:Splash0}) which runs
faster, and it has a proven topological significance. The main reason
$z_0$ is not the main character of this paper is that it is already
well known: restricted to knots $z_0$ is the Alexander polynomial and
considered on pure tangles (see Section~\ref{sec:g0}) it is equivalent to
the ``$\beta$-calculus'' of~\cite{Bar-NatanSelmani:MetaMonoids, KBH},
which in itself is mostly equivalent~\cite{Halacheva:Thesis} to Archibald's
calculus~\cite{Archibald:Thesis} and is a mild extension of the
Burau-Gassner theory of \cite{LeDimet:Gassner, KirkLivingstonWang:Gassner,
CimasoniTuraev:LagrangianRepresentation, CimasoniConway:BurauAlexander}.

As we shall see in Section~\ref{sec:g1}, $z_1$ arises from a certain Lie algebra $\frakg_1$, and as
we shall see in Section~\ref{sec:g0}, $z_0$ arises from a certain Lie algebra $\frakg_0$. Within
this introduction we only wish to define these two Lie algebras and explain how they arise as
``(solvable) approximations of $sl_2$''.

\begin{definition} Over the field $\bbQ$ of rational numbers, let
$\frakg_0$ be the 4-dimensional Lie algebra generated by elements $b$, $c$, $u$, and $w$,
such that $b$ is central, $[c,u]=u$, $[c,w]=-w$, and $[u,w]=b$:
\[ \frakg_0 \coloneqq \bbQ\langle b,c,u,w \rangle
  \left/\left(
    [b,-]=0,\, [c,u]=u,\, [c,w]=-w,\, [u,w]=b
  \right)\right..
\]
We grade $\frakg_0$ by declaring that $\deg(b,c,u,w)=(1,0,1,0)$.
\end{definition}

Note that $\frakg_0$ is solvable: Indeed
\[ \frakg_0\supset\langle b,u,w\rangle \supset \langle b \rangle \supset \{0\} \]
is a tower of ideals of $\frakg_0$ whose consequetive quotients are Abelian.

\begin{definition} Over the ring $R\coloneqq\bbQ[\epsilon]/(\epsilon^2=0)$, let $\frakg_1$
be the 4-dimensional Lie algebra generated by elements $b$, $c$, $u$, and $w$,
such that $b$ is central, $[c,u]=u$, $[c,w]=-w$, and $[u,w]=b-2\epsilon c$:
\[ \frakg_1 \coloneqq R\langle b,c,u,w \rangle
  \left/\left(
    [b,-]=0,\, [c,u]=u,\, [c,w]=-w,\, [u,w]=b-2\epsilon c
  \right)\right..
\]
We grade $R$ and $\frakg_1$ by declaring that $\deg(b,c,u,w,\epsilon)=(1,0,1,0,1)$.
\end{definition}

Note that over $\bbQ$, $\frakg_1$ is an 8-dimensional solvable Lie
algebra: Indeed
\[ \frakg_1
  \supset\langle b,u,w,\epsilon b,\epsilon c,\epsilon u,\epsilon w\rangle
  \supset \langle b,\epsilon b,\epsilon c,\epsilon u,\epsilon w\rangle
  \supset 0
\]
is a tower of ideals of $\frakg_1/\bbQ$ whose consequetive quotients are Abelian.

For the purposes of this paper, we do not need to know that $\frakg_0$
and $\frakg_1$ are related to the classical Lie algebra $sl_2$, and
hence the next few paragraphs\footnoteT{Trigger warning: Borel and Cartan
subalgebras, Lie bialgebras, Drinfel'd doubles.} can safely be skipped. Yet these
paragraphs say that ``there's more where $\frakg_0$ and $\frakg_1$ came from'', and
strongly suggest that there's more where our $z_0$ and $z_1$ come from.

Let $\frakg$ be an arbitrary semi-simple Lie algebra over $\bbQ$, and let
$\frakg=\frakn^+\oplus\frakh\oplus\frakn^-$ be

MORE.

\section{Algebras, Yang-Baxter Elements and Spinners, Invariants}
\label{sec:generalities}

MORE.

\section{The Lie Algebra $\frakg_0$, the Invariant $z_0$, and the Alexander Polynomial}
\label{sec:g0}

\begin{figure}
\begin{multicols}{2}
\noindent{\bf A Demo Program for $z_0$} \vskip 2mm
\par\noindent\includegraphics[width=\linewidth]{Snips/Program0.pdf}
\columnbreak
\par\noindent{\bf Initial Data} \vskip 2mm
\par\noindent\includegraphics[width=\linewidth]{Snips/Data0.pdf}
\par\noindent{\bf The Knot $8_{17}$} \vskip 2mm
\par\noindent\includegraphics[width=\linewidth]{Snips/Run0.pdf}
\captionsetup{width=0.9\linewidth}
\caption{
  A demo program computing $z_0$, some initial data, and a sample
  run on the knot $8_{17}$. The program is written in {\sl
  Mathematica}~\cite{Wolfram:Mathematica} and is available at~\cite{Self}.
}
\label{fig:Splash0}
\end{multicols}
\end{figure}

\section{The Lie Algebra $\frakg_1$ and the Invariant $z_1$}
\label{sec:g1}

MORE.

\section{Computing $z_1$}
\label{sec:z1}

MORE.

\section{An Aside on Ribbon Knots}
\label{sec:ribbon}

MORE.

\section{Questions}
\label{sec:questions}

MORE.

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