The Homflypt Yang–Baxter homology conjecture for the two-element rack

Let $R_{(m)}$ be the rack considered in the paper, let $H_n=H_n(R_{(m)})$ denote its $n$th Yang–Baxter homology over $k$, and specialize to $m=2$. Define

$$s_n=\sum_{i=1}^{n+1}f_i,$$

where $f_1=f_2=1$ and $(f_i)$ is the Fibonacci sequence. Let $a_n$ be determined by

$$2^n=2+a_{n-1}+s_{n-3}+a_n+s_{n-2},\qquad a_1=0.$$

The Homflypt Yang–Baxter homology conjecture. For $m=2$, one has

$$H_n=k^2\bigoplus \left(k/(1-y^2)\right)^{a_n}\bigoplus \left(k/(1-y^4)\right)^{s_{n-2}}.$$

The formula was verified by computer for $n\leq 11$, but remains conjectural in general. It describes the free part and the two types of torsion predicted for the Yang–Baxter homology of the two-element rack.