9 problems

The refined tangle equations are formulated for long knots with their longitude, using Gauss diagram formulas and weights associated with Vassiliev invariants. Generalization conje…

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…

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 $R_{(m)}$ 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

Let $K$ be a nontrivial prime alternating knot with crossing number $c(K)$, let $W_\pm(K,m)$ denote its positive or negative $m$-twisted Whitehead double, and let…

Let $K$ be a prime alternating knot with crossing number $c(K)$, let $W_\pm(K,m)$ denote its positive or negative $m$-twisted Whitehead double, and let $P_{W_\pm(K,m)}(v,z)$ be its…

HOMFLYPT quantum one-cocycle conjecture.

A colored braid is mapped to a complex of singular Soergel bimodules, categorifying the corresponding colored braid-group element. The closure of the braid has a colored HOMFLYPT p…