14 problems

Let $p$ be an odd prime, and let $\operatorname{R}_p$ be the dihedral quandle of order $p$. Write $\operatorname{H}^{\rm R}_n(\operatorname{R}_p)$ for its rack homology. Fenn–Rourk…

Combined injectivity conjecture. For odd $k$ and $n>1$, the direct-sum map

Let $R_k$ be an odd dihedral quandle, and let $(h_s)_*:H_n^R(R_k)\to H_{n+2}^R(R_k)$ be the homology operation induced by the degree-two chain map $h_s$. Injectivity conjecture for…

Chain-homotopy conjecture. The maps $g_1^j$ and $g_2^j$ are chain homotopic. The source notes that these maps are chain maps for the stated prime orders; proving their chain homoto…

Let $R_q$ be the dihedral quandle of odd prime order $q$, and let $H_n^R(R_q)$ denote its $n$-th rack homology group. Torsion-annihilation conjecture. For $n>1$, the torsion subgro…

Splitting conjecture. This long exact sequence should split into short exact sequences for every $n$.

Let $X$ be a quandle, and consider the long exact sequence in homology induced by the short exact sequence of chain complexes … This gives connecting homomorphisms…

Let $\operatorname{R}_{2k}$ be the dihedral quandle of order $2k$, and let $\operatorname{H}^{\rm R}_n(\operatorname{R}_{2k})$ denote its rack homology. Torsion-annihilator conject…

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

An associative shelf is a set equipped with an associative, self-distributive binary operation. Its two-term (rack) homology groups are the homology groups obtained from the associ…

Dihedral-quandle torsion conjecture. If $k\neq 2^t$ for $t>1$, then $k$ annihilates $\operatorname{tor} H_n^R(R_{2k})$. If $k=2^t$ with $t>1$, then $2k$ is the smallest positive in…

Inner-automorphism-order conjecture. The order $|G_Q^0|$ annihilates the torsion of $H_n^R(Q)$; equivalently,

Order-annihilation conjecture. The order $|Q|$ annihilates the torsion of $H_n^R(Q)$; equivalently,