Minimal crossing number implies minimal supporting genus
arXiv:2012.09000 · doi:10.1112/blms.12491
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.
10 pages, 3 figures. Comments welcome. Proof of Theorem 10 reorganised