Minimal crossing number implies minimal supporting genus

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