On Bar-Natan - van der Veen's perturbed Gaussians

TL;DR

This paper explores a new family of polynomial knot invariants based on Gaussian calculus, revealing their properties, relations to known invariants, and behavior under knot operations.

Contribution

It introduces and analyzes a novel family of polynomial knot invariants, proving a vanishing conjecture and providing explicit formulas for specific cases.

Findings

Half of the polynomials vanish as conjectured

Explicit formulas for three knot polynomial invariants

Behavior under connected sum of knots studied

Abstract

We elucidate further properties of the novel family of polynomial time knot polynomials devised by Bar-Natan and van der Veen based on the Gaussian calculus of generating series for noncommutative algebras. These polynomials determine all coloured Jones polynomials and the simplest of these is expected to coincide with the one-variable 2-loop polynomial. We prove a conjecture stating that half of these polynomials vanish and give concrete formulas for three of these knot polynomial invariants. We also study the behaviour of these polynomials under the connected sum of knots.