15 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…

Vanishing boundary conjecture. For every finite quandle $X$,

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…

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

Delayed Fibonacci conjecture. For $n>1$,

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

Rank-two conjecture. The group $H^2_{\rm TQ}(A;A)$ has rank at least two.

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…

Delayed-Fibonacci torsion conjecture. The torsion subgroup of $\operatorname{H}^{\rm Q}_n(X)$ is isomorphic to $\mathbb{Z}_4^{f_n}\oplus\mathbb{Z}_2^{f'_n}$. The displayed low-degr…

Let $GQ(o_1 \mid o_2 \mid \cdots \mid o_k)$ be the graphic quandle with $k$ orbits of sizes $o_1,o_2,\ldots,o_k$, respectively, with at least one orbit having more than two element…

Let $G={\cal M}_g$ be the mapping class group of the closed surface $\Sigma_g$, where $g\geq 2$, and let $X={\cal D}_g$ be the set of positive Dehn twists along unoriented simple c…

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,

Let $R_p$ be the rack under consideration, where $p$ is prime. The torsion subgroup of its homology is known to be $p$-primary. Niebur's exponent conjecture. The torsion in the hom…