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

Timothy Morris

Published 2019-02-18Version 1

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.