  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.
