The Jones-Krushkal polynomial and minimal diagrams of surface links

Homayun Karimi (McMaster University)

We prove a Kauffman-Murasugi-Thistlethwaite theorem for alternating
links in thickened surfaces. It implies 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.  This is joint work with Hans Boden.

3 thoughts on “The Jones-Krushkal polynomial and minimal diagrams of surface links

  1. Awesome! Thank you Homayun! If you would like to leave a question for Dr. Karimi, please reply to this comment below.

  2. There are further generalizations. In fact there is a universal “polynomial” invariant for
    knots in thickened surfaces, which lives in the skein module (see,
    along with some intermediate generalizations which will be appear in a future paper soon.

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