14 problems

Let $M$ be a hyperbolic knot complement in $\mathbb{S}^3$. Two finite-volume hyperbolic 3-manifolds are commensurable when they share a common finite-degree cover; the commensurabi…

Commensurability conjecture. Any two hyperbolic structures with the same length spectrum are commensurable.

Hoste–Shanahan's commensurability conjecture. The complement $\mathbb S^3-J(m,n)$ is commensurable to a fibered knot in a $\mathbb Z/2\mathbb Z$-homology sphere if and only if…

Commensurable-sides conjecture. The tiling must have commensurable sides.

Dodecahedral-knot conjecture. The only two knots with hidden symmetries are the two dodecahedral knots of Aitchison and Rubinstein.

Incommensurability conjecture. The following assertions hold:

Let $\mathbb{G}(\Gamma_1)$ and $\mathbb{G}(\Gamma_2)$ be $2$-dimensional right-angled Artin groups, meaning that their commutation graphs are triangle-free. Let…

Common finite cover conjecture. There exist finite-sheeted covers

Let $G$ be a Fuchsian group acting on the hyperbolic plane. The axes of the hyperbolic elements of $G$ form a set of geodesics, and the commensurability class of $G$ is the class o…

Polynomial-bound conjecture. The upper bound on $C^{\mathrm{arith}}_n(v)$ given in the theorem can be improved to a polynomial bound; that is, there exist constants $A,B>0$ such th…

Infinitude conjecture. There are infinitely many minimal combinatorial tetrahedral tessellations.

Let $M$ and $N$ be closed irreducible geometric 3-manifolds. Their abelian and $SU(2)$ Turaev–Viro invariants are the collections of invariants associated to abelian and $SU(2)$ th…

A commensurability class of hyperbolic $3$-manifolds is called rigid when its unique minimal element has a single cusp whose horospherical cross-section is a Euclidean turnover. Tw…

Riley–Wielenberg conjecture.