Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial

In this paper, we analyze the homology of the Yang-Baxter Operators $R_{(m)}$ yielding the HOMFLYPT polynomial, reducing the computation of the $n$-th homology of $R_{(m)}$ for arbitrary $m$ to the computation of $n+1$ initial conditions. We then produce the explicit formulas for the third and fourth homology.