Relating virtual knot invariants to links in ${\mathbb{S}}^3$

Abstract

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in ${\mathbb{S}}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J\bigsqcup K$ with $J$ a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links $J\bigsqcup K_1\bigsqcup K_2$ with $J$ a connected sum of trefoils or figure-eights. Our main technical tool is virtual covers. This technique, due to Manturov and the first author, associates a virtual knot $\upsilon$ to a link $J\bigsqcup K$, where $J$ is fibered and $\text{lk}(J,K)=0$. Here we extend virtual covers to all multicomponent links $L=J\bigsqcup K$, with $K$ a knot. It is shown that an unknotted component $J_0$ can be added to $L$ so that $J_0\bigsqcup J$ is fibered and $K$ has algebraic intersection number zero with a fiber of $J_0\bigsqcup J$. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots.

Improved main theorem and standardized orientation conventions. This version to appear in New York Journal of Mathematics

References in corpus (1)

• On Alexander-Conway Polynomials for Virtual Knots and Links