The computation of higher order Alexander invariants

TL;DR

The paper addresses the computational bottleneck in higher order Alexander invariants $\Delta_i(t)$ by exploiting the primary decomposition of the knot's Alexander module. It proves a main theorem: if every irreducible factor of the Alexander polynomial $\Delta(t)=\det(A_K)$ has exponent at most $6$, then each $\Delta_i(t)$ is determined by the ranks of $A_K$ over field extensions $\mathbb{Q}(\omega)$ and the $\mathrm{LCM}$ of the denominators of the reduced entries of $A_K^{-1}$. This yields dramatic speedups, enabling the computation of $\Delta_i(t)$ for thousands of knots up to $12$ crossings, tens of thousands up to $14$ crossings, and millions up to $16$ crossings in feasible times, with random high-crossing knots also tractable; the approach facilitates large-scale data generation for knot invariants and potential future extensions using Seifert matrices for even higher crossings.

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.

Theorems & Definitions (13)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Theorem 3
• Corollary 4
• proof
• Lemma 5
• proof
• Lemma 6