13 problems

A closed, orientable 3-manifold $M$ is Haken if it contains an incompressible embedded surface, and it is virtually Haken if it has a finite cover that is Haken. The virtually Hake…

Przytycki's torsion conjecture. The following are equivalent:

Let $M$ be a submanifold of a rational homology sphere. Assume that $M$ contains no closed, oriented incompressible surface. Let ${\cal S}_3(M)$ denote its Homflypt skein module, l…

Torus theorem. Either $M$ contains an incompressible torus, or $M$ is a small Seifert fibre space.

Homological detection conjecture. If

A $\pi_1$-injective surface in a $3$-manifold is a surface whose inclusion induces an injective homomorphism on fundamental groups. The fundamental group of a $3$-manifold $M$ is d…

Refined Slopes Conjecture. The Jones slopes of $K$ are boundary slopes, and the cluster points

Let $M=\Gamma\backslash G/K$ be a closed locally symmetric manifold, where $G$ is a semisimple Lie group with finite center, without compact factors and without factors locally iso…

Let $L$ be a non-trivial, non-split link, and let $E(L)$ denote its exterior. A closed surface $F$ containing $L$ has each component of $L$ non-separating in $F$ when no component…

Let $M$ be an oriented irreducible 3-manifold with $\pi_1(M)$ containing $\mathbb{Z}\oplus\mathbb{Z}$. Torus theorem conjecture. Then either $M$ contains an incompressible torus or…

Let $K$ be a prime non-torus knot, let $F$ be a non-orientable spanning surface for $K$, and let $E(K)$ be the exterior of $K$. The surface $F\cap E(K)$ is required to be geometric…

Let a geometrically incompressible surface in a filled knot space have restrictions to the knot space, and call the restrictions non-isotopic if they are not isotopic there. Polygo…

Incompressible-surface conjecture. Every closed or complete finite-volume hyperbolic 3-manifold admits an immersed or embedded closed incompressible surface.