On Virtual Crossing Numbers for Virtual Knots

TL;DR

This paper proves that for certain families of virtual knots, the minimal virtual crossing number increases quadratically with the minimal classical crossing number, surpassing previous linear estimates.

Contribution

It establishes a quadratic growth rate for virtual crossing numbers in some virtual knot families, challenging prior linear bounds.

Findings

Virtual crossing number can grow quadratically with classical crossings.

Previous estimates only showed linear growth.

No virtual knot was previously known with virtual crossing number exceeding classical crossings.

Abstract

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.