Myopic Tutte polynomials and Khovanov homology in $\mathbb{R}P^3$

Abstract

We present a "myopic" Tutte polynomial for graphs on $\mathbb{R}P^2$ which takes only nullhomologous spanning subgraphs as input. It recovers the generalized Krushkal polynomial and Drobotukhina's analogue of the Jones polynomial for alternating, nullhomologous links in $\mathbb{R}P^3$. We use this myopic Tutte polynomial to prove an analogue of the Kauffman-Murasugi-Thistlethwaite Theorem, relating the Jones polynomial of an alternating link to certain refinements of the crossing number. Finally, we construct a spanning tree model for the Khovanov homology of nullhomologous links, mirroring work by Champanerkar-Kofman and Wehrli for links in $S^3$. For alternating links, we use our model to prove that the Khovanov homology in $\mathbb{Z}/2\mathbb{Z}$ coefficients is determined entirely by the Jones polynomial and signatures of the link.