Meander diagrams of virtual knots

TL;DR

This paper extends the concept of meander diagrams to virtual knots, proving their universality, and introduces virtual k-arc crossing numbers, showing their equivalence to classical counterparts for classical knots.

Contribution

It generalizes meander diagrams to virtual knots and introduces virtual k-arc crossing numbers, establishing their relation to classical invariants.

Findings

Meander and semimeander diagrams are universal for virtual knots.

Virtual k-arc crossing numbers are introduced as new invariants.

For classical knots, virtual and classical k-arc crossing numbers are equal.

Abstract

For classical knots, there is a concept of (semi)meander diagrams; in this short note we generalize this concept to virtual knots and prove that the classes of meander and semimeander diagrams are universal (this was known for classical knots). We also introduce a new class of invariants for virtual knots -- virtual $k$-arc crossing numbers and we use Manturov projection to show that for all classical knots the virtual $k$-arc crossing number equals to the classical $k$-arc crossing number.