The Jones-Krushkal polynomial and minimal diagrams of surface links

We prove an analogue of the Kauffman-Murasugi-Thistlethwaite theorem for alternating links in 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. The proof holds more generally for links admitting adequate diagrams and the key ingredient is a two-variable generalization of the Jones polynomial for surface links defined by Krushkal. This result extends the first and second Tait conjectures to alternating links in thickened surfaces and also to alternating virtual links. This is joint work with Homayun Karimi.
Time permitting, we will discuss a new invariant of link in surfaces called the homotopy Kauffman bracket. We give several examples to show the homotopy Kauffman bracket is stronger than the homological Kauffman braket, and we use it to extend the first and second Tait conjectures to weakly reduced alternating diagrams in surfaces. This part is work in progress and is joint with Homayun Karimi and Adam Sikora.