Some non-abelian covers of knots with non-trivial Alexander polynomial

TL;DR

This paper constructs non-abelian covers of knots with non-trivial Alexander polynomial, providing bounds on their size and Betti number, and demonstrating the existence of non-peripheral homology in these covers.

Contribution

It introduces new bounds for non-cyclic covers of knots with non-trivial Alexander polynomial and analyzes their homological properties using classical and Alexander stratification techniques.

Findings

Bounds on the order of non-abelian covers in terms of crossing number

Betti number of covers can be arbitrarily large

Existence of non-peripheral homology in certain covers

Abstract

Let $K$ be a tame knot embedded in $\mathbf{S}^3$. We address the problem of finding the minimal degree non-cyclic cover $p:X \rightarrow \mathbf{S}^3 \smallsetminus K$. When $K$ has non-trivial Alexander polynomial we construct finite non-abelian representations $\rho:\pi_1\left(\mathbf{S}^3 \smallsetminus K\right) \rightarrow G$, and provide bounds for the order of $G$ in terms of the crossing number of $K$ which is an improvement on a result of Broaddus in this case. Using classical covering space theory along with the theory of Alexander stratifications we establish an upper and lower bound for the first betti number of the cover $X_\rho$ associated to the $\text{ker}(\rho)$ of $\mathbf{S}^3 \smallsetminus K$, consequently showing that it can be arbitrarily large. We also demonstrate that $X_\rho$ contains non-peripheral homology for certain computable examples, which mirrors a famous result of Cooper, Long, and Reid when $K$ is a knot with non-trivial Alexander polynomial.