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\def\title{Knot Invariants from Finite Dimensional Integration}

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\def\Abstract{{\raisebox{2mm}{\parbox[t]{3.96in}{ {\red\bf Abstract.}
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
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.

{\bf Q.} Are there any such things?
\newline{\bf A.} Yes.

\hangindent=5mm\hangafter=2
{\bf Q.} Are they any good?
\newline{\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\Knots{{\raisebox{0mm}{\parbox[t]{3.96in}{
{\bf The Good.} 1. At the centre of low dimensional topology.
\newline 2. ``Invariants'' connect to pretty much all of algebra.
\par{\bf The Agony.} 1\&2 don't talk to each other.
\begin{itemize}
\item Not enough topological applications for all these invariants.
\item The fancy algebra doesn't arise naturally within topology.
\end{itemize}
$\Longrightarrow$ We're still missing something about the relationship between knots and
algebra.
}}}}

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\begin{document} \latintext
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{\red\bf (Alternative) Gausssian Integration.}

{\bf Goal.} Compute
\[ I_1(0) \coloneqq \int d^nx P(x)\exp\left(-\frac12a^{ij}x_ix_j + V(x)\right). \]

{\bf Solution.} Set
\[
  I_\lambda(x) \coloneqq
    \int d^nx' P(x+x')\exp\left(-\frac{1}{2\lambda}a^{ij}x'_ix'_j + V(x+x')\right).
\]
Then $I_1(0)$ is what we want, $I_0(x) = (\det A)^{-1/2}P(x)\exp V(x)$, and
\[ \partial_\lambda I_\lambda(x) =
  \frac{1}{2\lambda^2}
    \int d^nx'a^{ij}x'_ix'_jP(x+x')\exp\left(-\frac{1}{2\lambda}a^{ij}x'_ix'_j + V(x+x')\right)
\]
While with $g_{ij}$ the inverse matrix of $a^{ij}$,
\begin{multline*}
  \frac12g_{ij}\partial_{x_i}\partial_{x_j}I_\lambda(x)
  = \int d^nx' \frac12g_{ij}(\partial_{x_i}-\partial_{x'_i})(\partial_{x_j}-\partial_{x'_j})
    P(x+x')\exp\left(-\frac{1}{2\lambda}a^{ij}x'_ix'_j + V(x+x')\right) \\
  = \frac{1}{2\lambda^2}
    \int d^nx'a^{ij}x'_ix'_jP(x+x')\exp\left(-\frac{1}{2\lambda}a^{ij}x'_ix'_j + V(x+x')\right).
\end{multline*}
Hence
\[ \partial_\lambda I_\lambda(x) = \frac12g_{ij}\partial_{x_i}\partial_{x_j}I_\lambda(x), \]
and therefore
\[ I_\lambda(x) = (\det A)^{-1/2}
  \exp\left(\frac{\lambda}{2}g_{ij}\partial_{x_i}\partial_{x_j}\right) P(x)\exp V(x).
\]

\begin{multicols*}{2}

\par{\red\bf References.}
\par\vspace{-3mm}
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\vskip -3mm\rule{\linewidth}{1pt}\vspace{0mm}

{\bf\red Acknowledgement.} This work was partially supported by NSERC
grant RGPIN-2018-04350 and by the Chu Family Foundation (NYC).

\end{multicols*}

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