12 problems

Filtered-subcomplex splitting conjecture. Over $k=\mathbb{Z}[y^2]$,

Let $R_{(m)}$ be the level-$m$ Yang–Baxter operator and let $k=\mathbb{Z}[y^2]$. Cyclic-decomposition conjecture. The homology $H_n(R_{(m)})$ can be decomposed as a direct sum of c…

Let $R_{(m)}$ be the level-$m$ Yang–Baxter operator yielding the HOMFLYPT polynomial, let $k=\mathbb{Z}[y^2]$, and let $a_6(m),b_6(m),c_6(m),d_6(m)$ denote the multiplicities in th…

Fix a decomposition of the alphabet $X_{(m)}=A\cup B$ and let $C_\bullet^{(m,A,B)}$ be the subchain complex generated by words that begin with letters from $A$ and end with letters…

Let $R_{(m)}$ be the level-$m$ Yang–Baxter operator yielding the HOMFLYPT polynomial, and let $k=\mathbb{Z}[y^2]$. The notation $\binom{m-1}{j}$ denotes a binomial coefficient. Fif…

Let $R_{(2)}$ be the level-two Yang–Baxter operator, let $H_n(R_{(2)})$ denote its $n$th homology, and let $k$ be the coefficient ring. Let $f_n$ be the Fibonacci numbers, with…

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…

Przytycki–Vojtěchovský conjecture.

Let $R_{(m)}$ be the rack considered in the paper, let $H_3(R_{(m)})$ denote its third Yang–Baxter homology over $k$, and let $m$ be the parameter indexing this rack family. The th…

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

Yang–Baxter homology conjecture. The homology groups satisfy

Homology decomposition conjecture. For every $n$,