59 problems

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 $L$ be an oriented link in the $3$-sphere, and let its $n$-quandle be the quandle obtained from the link quandle by imposing the $n$-quandle relations. Let $M_n(L)$ denote the…

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 $R_3$ be the three-element dihedral quandle, and consider the cocycle invariant of an oriented knot and of its mirror image with the same orientation, using the same cocycle an…

Let $R_3$ be the three-element dihedral quandle, and let $q$ specify the coefficient group $(\mathbb{Z}_q)^3$ in the wreath-product coefficient action. A knot is colored by $R_3$ t…

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 $K$ be a link. Let $A_p(K)$ be its Alexander module over $\Lambda_p=\mathbb{F}_p[t,t^{-1}]$, let $h=t^2+\kappa t+1$ be the irreducible quadratic defining the quadratic quandle,…

Delayed Fibonacci conjecture. The quandle homology of $R_k$ satisfies

Hayashi's conjecture. For every connected quandle with this profile, $l_i$ divides $l_t$ for every $1\leq i\leq t$; equivalently, all right translation maps are regular permutation…

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 $n\geq 3$ be an odd integer, and set $k:=2n-1$. Let $\operatorname{Good}\Lambda_{4n,k}$ denote the set of good involutions associated with the twisted conjugation subquandle in…

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 $G$ 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 $X_1,X_2,\ldots,X_k$ be quandles, with associated cocycles $\phi_1,\ldots,\phi_k$ and cocycles $\psi_1,\ldots,\psi_k$ on the idempotent quandles…

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…

A profinite quandle is a quandle that is an inverse limit of finite quandles. For a quandle $Q$, let $coperatorname{L}(Q)$ denote its lattice of subquandles. Complementation conjec…

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…