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Knots and Representation Theory
April 30, 2013 16:30, Moscow

Virtual Knots and Fibered Knots

M. Chrisman

Monmouth University

Abstract: Let $L=J \sqcup K$ be a two component link in $S^3$ such that $J$ is a fibered knot and the linking number of $J$ and $K$ is zero. Let $\mathscr{FL}$ denote the ambient isotopy classes of such links $L$ and let $\mathscr{VK}$ denote the set of virtual isotopy classes of virtual knots. We construct a surjective map $\Gamma: \mathscr{FL} \to \mathscr{VK}$. The map is used to give applications of virtual knot theory to classical knot theory. We use virtual knot invariants to distinguish classical two component links, detect non-invertibility of two component classical links, and establish minimality theorems for diagrams of classical two component links. The examples reveal that subtle geometric properties of classical knots can be detected easily using virtual knot theory.