Skein theoretic approach to Yang-Baxter homology

Abstract

We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB operator R for Jones, normalized for homology, admits a skein decomposition R=I+βα, where α:V⊗2→k is a “cup” pairing map and β:k→V⊗2 is a “cap” copairing map, and differentials in the chain complex associated to R can be decomposed into horizontal tensor concatenations of cups and caps. We apply our skein theoretic approach to determine the second and third YB homology groups, confirming a conjecture of Przytycki and Wang. Further, we compute the cohomology groups of R, and provide computations in higher dimensions that yield some annihilations of submodules.