# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.07756