Minimal crossing number implies minimal supporting genus

Hans U. Boden; William Rushworth

doi:10.1112/blms.12491

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arXiv · 2012.09000

Minimal crossing number implies minimal supporting genus

Hans U. Boden·William Rushworth

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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.

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Hans U. Boden, William Rushworth. 2020-12-16. Minimal crossing number implies minimal supporting genus. https://doi.org/10.1112/blms.12491

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