The computation of higher order Alexander invariants

TL;DR

This paper develops efficient methods to compute higher order Alexander invariants for knots, overcoming computational challenges of traditional Smith normal form algorithms, enabling analysis of knots with up to 100 crossings.

Contribution

The paper introduces novel algorithms that significantly improve the computation of higher order Alexander invariants for complex knots.

Findings

Effective computation of Alexander polynomials for knots with up to 100 crossings

Overcomes limitations of Smith normal form algorithms

Enables analysis of more complex knot invariants

Abstract

In 1928, Alexander defined a sequence of knot polynomials, D_i(K). The first, D_1(K), is the classical Alexander polynomial. These are easily defined in terms of the homology of the infinite cyclic cover of the knot. In theory they can be computed by putting an associated Alexander matrix in Smith normal form. However, standard algorithms for computing the Smith form become impractically slow, even for some 16 crossing knots. Here, methods are developed that can effectively compute the Alexander polynomials of knots with up to 100 crossings.