A perturbed Alexander polynomial for singular knots

TL;DR

This paper introduces a perturbed Alexander polynomial for singular knots and tangles, demonstrating its enhancement over previous versions and establishing its consistency with existing definitions for singular long knots.

Contribution

It presents a new perturbed Alexander polynomial for singular knots and tangles, significantly strengthening the invariant and confirming its agreement with prior definitions for singular long knots.

Findings

The perturbed Alexander polynomial is a stronger invariant.

It agrees with existing definitions for singular long knots.

The method extends the classical Alexander polynomial to singular cases.

Abstract

We introduce a version of the Alexander polynomial for singular knots and tangles and show how it can be strengthened considerably by introducing a perturbation. For singular long knots, we also prove that our Alexander polynomial agrees with previous definitions.