On computing higher-order Alexander modules of knots

TL;DR

This paper introduces an algorithm to compute higher-order Alexander modules of knots, enabling better genus bounds and mutation detection, thus advancing knot invariants analysis.

Contribution

It presents the first algorithm for calculating the first-order Alexander module for any knot, improving knot invariant tools.

Findings

Higher-order Alexander polynomials give better genus bounds.

They can detect knot mutation.

The algorithm solves the word problem in finitely presented Z[Z]-modules.

Abstract

Cochran defined the nth-order integral Alexander module of a knot in the three sphere as the first homology group of the knot's (n+1)th-iterated abelian cover. The case n=0 gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain, from which one can extract a higher-order Alexander polynomial. We present an algorithm to compute the first-order Alexander module for any knot. As applications, we show that these higher-order Alexander polynomials provide a better bound on the knot genus than does the classical Alexander polynomial, and that they detect mutation. Included in this algorithm is a solution to the word problem in finitely presented Z[Z]-modules.