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 $R = I + \beta\alpha$, where $\alpha: V^{\otimes 2} \rightarrow k$ is a "cup" pairing map and $\beta: k \rightarrow V^{\otimes 2}$ 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.