Tensors, Gaussians and the Alexander Polynomial

TL;DR

This paper introduces a Gaussian-based model leveraging the Heisenberg algebra and tensor contractions to compute the Alexander polynomial of knots, connecting knot invariants with Gaussian functions and algebraic structures.

Contribution

It develops a novel Gaussian model that uses tensor contraction and algebraic methods to compute the Alexander polynomial of knots.

Findings

Partition function of the Gaussian recovers the Alexander polynomial

Uses a presentation matrix as a precision matrix in the Gaussian model

Builds on the approach of Bar-Natan and Van der Veen for universal knot invariants

Abstract

Building on the approach of Bar-Natan and Van der Veen to universal knot invariants using (perturbed) Gaussian functions, we develop a Gaussian model to compute the Alexander polynomial $\Delta_{\mathcal{K}}(T)$ of an oriented knot $\mathcal{K}$ in $S^3$. Using the Heisenberg algebra and a tensor-contraction formalism, we associate to a knot a Gaussian function whose partition function recovers $\Delta_{\mathcal K}(T)$. Here, a presentation matrix of the Alexander module plays the role of a precision matrix of the Gaussian function.