A Perturbed-Alexander Invariant

TL;DR

This paper introduces a simplified and efficient computational method for a known knot invariant, the $ ho_1$, using inverse matrices and quadratic expressions, potentially revealing new topological insights.

Contribution

It provides the simplest and fastest formulas for computing the $ ho_1$ invariant, enhancing computational efficiency and suggesting a topological interpretation.

Findings

Developed concise formulas for $ ho_1$ invariant

Created a fast computer program for invariant calculation

Linked the formulas to the Alexander polynomial

Abstract

In this note we give concise formulas, which lead to a simple and fast computer program that computes a powerful knot invariant. This invariant $\rho_1$ is not new, yet our formulas are by far the simplest and fastest: given a knot we write one of the standard matrices $A$ whose determinant is its Alexander polynomial, yet instead of computing the determinant we consider a certain quadratic expression in the entries of $A^{-1}$. The proximity of our formulas to the Alexander polynomial suggest that they should have a topological explanation. This we don't have yet.