11 problems

Nullhomotopic knot complement conjecture. If $K_1$ and $K_2$ have orientation-preservingly homeomorphic exteriors, then there exists an orientation-preserving homeomorphism of $Y$…

Knot complement conjecture. If $K_1$ and $K_2$ have orientation-preservingly homeomorphic exteriors, then there exists an orientation-preserving homeomorphism of $Y$ mapping $K_1$…

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…

For $R>0$, consider hyperbolic knot complements in $\mathbb{S}^3$ with hidden symmetries and volume less than $R$. Finiteness conjecture. For any $R>0$, at most finitely many hyper…

Assume that $Y=S^3\setminus\mathcal{K}$ is an ideally triangulated knot complement. A link $L\subset S^3\setminus\mathcal{K}$ is even if it represents the zero homology class in ……

Rigid Cusp Conjecture. If a hyperbolic knot complement $\mathbb{S}^3\setminus K$ covers an orbifold with a rigid cusp, then $\mathbb{S}^3\setminus K$ covers an orbifold with an…

Finiteness conjecture. For any $R>0$, at most finitely many hyperbolic knot complements have hidden symmetries and volume less than $R$.

Let $M$ be a closed oriented $3$-manifold, and let $K_1$ and $K_2$ be knots in $M$ whose complements are homeomorphic by an orientation preserving homeomorphism and are not homeomo…

Splicing conjecture. The manifold $Y$ is an $L$-space if and only if all of the following conditions hold:

Let $K$ be an L-space knot, meaning a knot for which $S^3_n(K)$ is an L-space for some $n>0$. An essential Conway sphere is an essential $4$-punctured sphere in the knot complement…

Topological invariance conjecture. The construction of $\hat{\mathcal I}_M$ produces a topological invariant of $M$, independent of the triangulation and of any other choices made.