An Unknotting Index for Virtual Links

TL;DR

This paper introduces an unknotting index for virtual links, providing bounds based on diagram span, linking number, and warping degree, to measure how virtual links can be simplified to trivial links.

Contribution

It defines the unknotting index for virtual links and establishes bounds using diagram invariants, applying these to virtualized pretzel links for the first time.

Findings

Lower bounds derived from span and linking number

Upper bounds obtained via warping degree

Applied bounds to virtualized pretzel links

Abstract

Given a virtual link diagram $D$, we define its unknotting index $U(D)$ to be minimum among $(m, n)$ tuples, where $m$ stands for the number of crossings virtualized and $n$ stands for the number of classical crossing changes, to obtain a trivial link diagram. By using span of a diagram and linking number of a diagram we provide a lower bound for unknotting index of a virtual link. Then using warping degree of a diagram, we obtain an upper bound. Both these bounds are applied to find unknotting index for virtual links obtained from pretzel links by virtualizing some crossings