Tensors, Gaussians and the Alexander Polynomial

Abstract

In this paper, we formally show that the Alexander polynomial of a knot can be computed through contractions of a Gaussian function. We build on the approach of Bar-Natan and Van der Veen to universal knot invariants using (perturbed) Gaussian functions. We endow the Weyl--Heisenberg algebra with an XC-structure. Given a knot, the universal invariant arising from the XC-algebra induces a Gaussian function whose partition function recovers the Alexander polynomial. This formalism allows for the addition of suitable perturbations to the Gaussian function, from which perturbative expansions of knot invariants can be derived.

20 pages, 2 figures, v2: fixed typos, added XC-algebra axiom, v3: improved proof theorem 4.6, v4: improved wording, processed feedback committee, added example and appendix