Minimal crossing number implies minimal supporting genus

Abstract

A virtual link may be de ned 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. is 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. is extends corresponding results in the case of virtual knots due to Manturov and Chernov.