The Jones-Krushkal polynomial and minimal diagrams of surface links

Abstract

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating links in thickened surfaces. It states that any reduced alternating diagram of a link in a thickened surface has minimal crossing number, and any two reduced alternating diagrams of the same link have the same writhe. This result is proved more generally for link diagrams that are adequate, and the proof involves a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. The main result is used to establish the first and second Tait conjectures for links in thickened surfaces and for virtual links.

32 pages, 20 figures, and 1 table

References in corpus (2)

• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• A characterization of alternating links in thickened surfaces

Cited by in corpus (6)

• Minimal crossing number implies minimal supporting genus
• Adequate links in thickened surfaces and the generalized Tait conjectures
• The Gordon-Litherland pairing for links in thickened surfaces
• Homological Polynomial Coefficients and the Twist Number of Alternating Surface Links
• A volumish theorem for alternating virtual links
• The homological arrow polynomial for virtual links