10 problems

Let $K$ be a hyperbolic knot, and consider the commensurability class of its complement $S^3\setminus K$. A knot complement is a complement of a knot in $S^3$ that lies in this cla…

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.

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…

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 $M$ 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.