36 problems

Let $Q$ and $Q'$ be symplectic quandles of the same dimension over the ring $\mathbb{Z}_n$. They are isomorphism–isometry conjecture. Isomorphic as quandles if and only if they are…

Let $Q$ be a finite quandle, and let $qp_Q(s,t)$ denote its quandle polynomial. A quandle is Latin when its operation is left-invertible, in addition to the right-invertibility req…

Let $K$ be a link. Let $A_p(K)$ be its Alexander module, let $A'_p(K)$ be the submodule of elements annihilated by $h$, and let $\operatorname{Rep}'(K,\mathbb{E}_p)$ be the annihil…

Let $X$ be a finite latin quandle, and let $u\in\mathbb{Z}[X]$ be a nonzero idempotent, so $u^2=u$. An element of $X$ is called a trivial idempotent, and $\varepsilon$ denotes the…

Conjugation quandle detection conjecture. The collection of all conjugation quandles over all dihedral groups is able to detect causality across $(2+1)$ globally hyperbolic spaceti…

Let ${Xi}{i i I}$ be a family of quandles, and suppose that each integral quandle ring $\mathbb{Z}[Xi]$ has only trivial idempotents. Let $\star{i i I}Xi$ denote their free product q…

For a quandle $X$, its integral quandle ring is $\mathbb{Z}[X]$, and an idempotent is an element $e$ satisfying $e^2=e$. The idempotents $0$ and the basis elements of $X$ are calle…

Let $\mathbb{k}$ be an integral domain with unity and let $X$ be a semi-latin quandle, meaning a quandle whose left and right translations satisfy the relevant latin-type condition…

A quandle is an algebraic structure with its standard quandle operation, and a quandle is residually finite if distinct elements can be separated by homomorphisms to finite quandle…

Let $G$ be a group, let $X$ be a subset of $G$, and let $H:=\langle X\rangle\leq G$. Assume that $X$ is a conjugacy class of $H$. Central-translate conjecture. If there exist disti…

A quandle is a set equipped with a binary operation satisfying the quandle axioms; it is connected when its inner automorphism group acts transitively, involutory when each right t…

Let $Q=(Q,\star)$ be a quandle, meaning that $a\star a=a$ for every $a\in Q$ and every left translation $L_a\colon Q\to Q$, $x\mapsto a\star x$, is an automorphism. The left multip…

Let $X$ be a quandle and let $\phi:X\times X\to A$ be a $2$-cocycle. Let … be the set of idempotents in the quandle ring $\mathbb{Z}_2[X]$, assume that $Y$ is a quandle, and let…

Solvability and abelianness conjectures. (i) Every finite superconnected quandle is solvable. (ii) Every finite simple superconnected quandle is abelian.

Let $T$ be a symplectic quandle, and let the Allen–Swenberg series be the infinite family of links considered in the source, with $H$ the connected sum of two Hopf links. The sympl…

Let $L$ be a quasi-alternating link, and let $IQ(L)$ denote its involutory quandle. A quandle is left-orderable if it admits a strict total order invariant under all left translati…

Non-isomorphism conjecture. The quandles $d4ac_n(Q)$ and $d4ac_m(Q)$ are not isomorphic whenever $n\neq m$ and $n,m\in\mathbb{Z}^+$.

Let $R_n$ be a dihedral quandle of odd order $n$, let $\mathbb{C}[R_n]$ be its complex quandle algebra, and let $\sigma(\mathbb{C}[R_n])$ denote its Peirce spectrum. Peirce-spectru…

Let $\mathbb{R}_n=\{a_0,a_1,\cdots,a_{n-1}\}$ denote the dihedral quandle of order $n$, and let $\Delta^k(\mathbb{R}_n)$ denote the $k$th power of its augmentation ideal. Bardakov–…

Let $X$ be a quandle and let $\mathbb{Z}[X]$ denote its integral quandle ring. A quandle is semi-latin when every right multiplication restricted to each orbit is injective; a fini…

Superconnected quandle conjectures.

A finite rack $X$ has profile $1^{m_0}l_1^{m_1}l_2^{m_2}\cdots l_k^{m_k}$ when its inner automorphism has $m_0$ cycles of length $1$, $m_1$ cycles of length $l_1$, and so on. Hayas…

Enumeration conjecture.

BPS conjecture. begin{enumerate} item If $n>1$ is an odd integer, then

For a knot or link $L$, let $\mathcal{K}(L)$ denote its fundamental quandle, and let $\chi_L(X):=|\operatorname{Hom}_{\operatorname{quandle}}(\mathcal{K}(L),X)|$ be the quandle cou…