# The Jones-Krushkal polynomial and minimal diagrams of surface links

## TL;DR

This paper extends classical link invariants and minimal diagram properties to links in thickened surfaces and virtual links, proving analogues of Tait conjectures using a generalized Jones polynomial.

## Contribution

It introduces a two-variable Jones polynomial for surface links and proves minimal crossing number and writhe invariance for alternating diagrams in thickened surfaces.

## Key findings

- Proves a Kauffman-Murasugi-Thistlethwaite theorem for surface links.
- Establishes minimal crossing number for reduced alternating diagrams.
- Confirms Tait conjectures for links in thickened surfaces and virtual 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.