Fibered Knots and Virtual Knots

TL;DR

The paper introduces virtual covers as a new method to study classical knots using virtual knot theory, especially when related to fibered knots, providing examples and foundational theory.

Contribution

It develops the concept of virtual covers for classical knots, linking virtual knot theory with classical knot invariants and properties, particularly in the context of fibered knots.

Findings

Virtual covers can be constructed for classical knots relative to fibered knots.

The method reveals new insights into classical knots through virtual knot theory.

Several examples demonstrate the applicability of virtual covers to classical knot problems.

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 $\pi_J: \Sigma \times (0,1) \to \bar{S^3\backslash V(J)}$, where $V(J)$ is a regular neighborhood of $J$ satisfying $V(J) \cap \text{im}(K)=\emptyset$ and $\Sigma$ is a connected compact orientable 2-manifold. Let $K'$ be a knot in $\Sigma \times (0,1)$ such that $\pi_J(K')=K$. Then $K'$ 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.