A perturbed-Alexander invariant

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 suggests that they should have a topological explanation, which we do not have yet.