Minimal crossing number implies minimal supporting genus

TL;DR

This paper proves that minimal crossing diagrams of virtual links also have minimal supporting genus, introducing a new parity theory and extending classical link invariants to virtual links.

Contribution

It introduces a novel parity theory for virtual links and demonstrates that minimal crossing diagrams correspond to minimal genus, extending classical invariants to virtual links.

Findings

Minimal crossing virtual link diagrams have minimal genus.

Crossing, bridge, and ascending numbers do not decrease when classical links are viewed virtually.

Introduces a new parity theory for virtual links.

Abstract

A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.