Minimal crossing number implies minimal supporting genus

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.

Local transformations and functorial maps

“…Then we get the move H(2). The move H (2) produces the moves H(2) o , H(3) and H(3) o . Hence, the map f τ is invariant in the destination knot theory M ′ = {H(2)}.…”

“…Manturov in [23] are picture-valued invariants of knots. Parity brackets and functorial map allow one to establish minimality of knot diagrams and to construct (counter) examples [2,6,24].…”

Adequate links in thickened surfaces and the generalized Tait conjectures

“…The dual state S 0_ to S 0 differs from S _ at x only. The states S _ and S 0_ are adjacent in the cube of resolutions of D. Thus, (10) t .S 0_ / 1 Ä t .S _ / Ä t .S 0_ / C 1:…”

“…By Lemma 29, there are the two possibilities. If r.D 0 / D r.D/, then (10) and the inductive assumption imply that t .S / C t .S _ / Ä t.S 0 / C t .…”