Tensors, Gaussians and the Alexander Polynomial

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 $Δ_{\mathcal{K}}(T)$ 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 $Δ_{\mathcal K}(T)$. Here, a presentation matrix of the Alexander module plays the role of a precision matrix of the Gaussian function.