Minimal crossing number implies minimal supporting genus

Boden, Hans; Rushworth, William

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.

Citation

Boden, H., & Rushworth, W. (2021). Minimal crossing number implies minimal supporting genus. Bulletin of the London Mathematical Society, 53(4), 1174-1184. https://doi.org/10.1112/blms.12491

• Journal Article Type: Article
• Acceptance Date: Jan 11, 2021
• Online Publication Date: Apr 15, 2021
• Publication Date: 2021-08
• Deposit Date: Nov 19, 2022
• Journal: Bulletin of the London Mathematical Society
• Print ISSN: 0024-6093
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 53
• Issue: 4
• Pages: 1174-1184
• DOI: https://doi.org/10.1112/blms.12491
• Keywords: Geometric Topology
• Public URL: https://hull-repository.worktribe.com/output/4127508
• Related Public URLs: https://arxiv.org/abs/2012.09000