Fibered Knots and Virtual Knots

Abstract

We introduce a new technique for studying classical knots with the methods of virtual knot theory. Let $K$ be a knot and $J$ a knot in the complement of $K$ with $\text{lk}(J,K)=0$. Suppose there is covering space ${\pi }_J:\Sigma \times (0,1)\to \overline{S^3\backslash V(J)}$, where $V(J)$ is a regular neighborhood of $J$ satisfying $V(J)\cap \text{im}(K)=\varnothing$ and $\Sigma$ is a connected compact orientable 2-manifold. Let $K^{\prime }$ be a knot in $\Sigma \times (0,1)$ such that ${\pi }_J(K^{\prime })=K$. Then $K^{\prime }$ stabilizes to a virtual knot $\hat{K}$, called a virtual cover of $K$ relative to $J$. We investigate what can be said about a classical knot from its virtual covers in the case that $J$ is a fibered knot. Several examples and applications to classical knots are presented. A basic theory of virtual covers is established.

Improved exposition; added references