Virtual Covers of Links

TL;DR

This paper employs virtual knot theory to identify non-invertibility in classical links within three-dimensional space, leveraging virtual covers associated with fibered and virtually fibered links to detect symmetries.

Contribution

It introduces a novel application of virtual knot theory to analyze the invertibility of classical links via virtual covers, especially for links related to fibered and virtually fibered links.

Findings

Virtual covers can detect non-invertibility of classical links.

Symmetry conditions on virtual knots relate to link invertibility.

Virtual covers extend to links with virtually fibered components.

Abstract

We use virtual knot theory to detect the non-invertibility of some classical links in $\mathbb{S}^3$. These links appear in the study of virtual covers. Briefly, a virtual cover associates a virtual knot $\upsilon$ to a knot $K$ in a $3$-manifold $N$, under certain hypotheses on $K$ and $N$. Virtual covers of links in $\mathbb{S}^3$ come from taking $K$ to be in the complement $N$ of a fibered link $J$. If $J \sqcup K$ is invertible and $K$ is "close to" a fiber of $J$, then $\upsilon$ satisfies a symmetry condition to which some virtual knot polynomials are sensitive. We also discuss virtual covers of links $J \sqcup K$ where $J$ is not fibered but is virtually fibered (in the sense of W. Thurston).