Explicit special covers of alternating links

TL;DR

This paper constructs explicit special covers of prime alternating link complements with degree bounds related to crossing numbers, leading to new insights into their group properties and residual finiteness.

Contribution

It introduces a method to build special covers with degree bounds based on crossing numbers, enabling embedding into right-angled Artin and Coxeter groups.

Findings

Subgroups of link groups embed into right-angled Artin and Coxeter groups.

Provides bounds on the degree of covers related to crossing numbers.

Quantifies residual finiteness and Betti number growth in covers.

Abstract

Given a prime, alternating link diagram, we build a special cover of the link complement whose degree is bounded by a factorial function of the crossing number. It follows that a subgroup of the link group of that index embeds into right-angled Artin and Coxeter groups. Corollaries of this result include a quantification of residual finiteness, control of the growth of Betti numbers in covers, and an explicit bound on the rank of a Z-module on which the link group acts faithfully.