  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.
