12 problems

Let $R_{2n+1}$ denote the dihedral quandle of odd order $2n+1$, let $\Delta(R_{2n+1})$ be its fundamental ideal, and let $\Delta^{2n+1}(R_{2n+1})$ be its $(2n+1)$-st power. Let…

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…

Let $R_k$ be an odd dihedral quandle, and let $(h_s)_*$ and $(h'_{(s,0)})_*$ be the induced operations … … Here $h'_{(s,0)}=h'_0h_s$. Combined quandle injectivity conjecture. For o…

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…

Injectivity conjecture for $h_s$. For $n>1$, the homomorphism

Delayed Fibonacci conjecture. For $n>1$,

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 $\operatorname{R}_n$ be the dihedral quandle and let $R$ be a ring. For the quandle ring $R[\operatorname{R}_n]$, let $\Delta(\operatorname{R}_n)$ denote its augmentation ideal…

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…