On Virtual Crossing Number Estimates For Virtual Links

TL;DR

This paper investigates methods to estimate the minimal number of virtual crossings in virtual link diagrams using the $\xi$-polynomial, providing conditions for minimality, new examples, and open questions.

Contribution

It introduces the $\xi$-polynomial as a tool for estimating virtual crossing numbers and offers new sufficient conditions and examples for minimal virtual diagrams.

Findings

$\xi$-polynomial's leading degree estimates virtual crossing number

Provided sufficient conditions for diagram minimality

Presented new infinite series of minimal virtual link examples

Abstract

We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in $R^{2}$. We tackle this problem by the so-called $\xi$-polynomial whose leading (lowest) degree naturally estimates the virtual crossing number. Several sufficient conditions for minimality together with infinite series of new examples are given. We also state several open questions about $M$-diagrams, which are minimal according to our sufficient conditions.