Free-rank conjecture for Yang–Baxter homology

Let $R_{(m)}$ be the Yang–Baxter operator at level $m$, let $C_\bullet^{mf}$ and $C_\bullet^m$ be the associated chain complexes, and write $\operatorname{free\,rank}$ for the rank of the free part of homology. Free-rank conjecture.

$$\operatorname{free\,rank} H_n(C_\bullet^{mf}) = \begin{cases} 1 & m \leq n+1 \\ 0 & m>n+1. \end{cases}$$

Consequently, the free rank of $H_n(C_\bullet^m)$ should be $\sum_{k=0}^n\binom{m-1}{k}$ when $m\geq n+1$, and $2^{m-1}$ when $m\leq n+1$. This is proposed as a guide to further computations of the homology of the Yang–Baxter operators yielding the HOMFLYPT polynomial.