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

TL;DR

This paper explores the connection between virtual knot invariants and classical link invariants in three-dimensional space, introducing new techniques like fiber stabilization to extend the applicability of virtual covers to more complex links.

Contribution

It extends the virtual covers technique to all multicomponent links and relates virtual knot invariants to classical link invariants in $ ext{S}^3$, providing new insights into virtual knot theory.

Findings

Alexander polynomials of almost classical knots are specializations of multi-variable Alexander polynomials.

The index of a crossing relates to the Milnor triple linking number.

Fiber stabilization allows studying all links with virtual knots.

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 $J \sqcup K$ 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 $J \sqcup K_1 \sqcup K_2$ 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 $J \sqcup K$, where $J$ is fibered and $\text{lk}(J,K)=0$. Here we extend virtual covers to all multicomponent links $L=J \sqcup K$, with $K$ a knot. It is shown that an unknotted component $J_0$ can be added to $L$ so that $J_0 \sqcup J$ is fibered and $K$ has algebraic intersection number zero with a fiber of $J_0 \sqcup J$. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots.