16 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…

Let $K\subset\mathbb{S}^3$ be a hyperbolic knot, and let $\mathbb{S}^3\setminus K$ denote its complement. The complement has hidden symmetries when a homeomorphism between finite-d…

Let $K$ be a knot, let $G=SU(N)$, and consider the totally symmetric representation. Write … and let $\alpha_{-k-1}(K;N,q)$ denote the coefficients appearing in the inverted Habiro…

Let $M$ be a hyperbolic knot complement in $\mathbb{S}^3$ with hidden symmetries. A cusp of an orbifold is a $(2,3,6)$-cusp when its cross-section is the Euclidean triangle orbifol…

Quantum A-polynomial annihilation conjecture. The series $F_K(x,q)$ is annihilated by the quantum $A$-polynomial:

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 ……

Let $N$ be an oriented, closed $3$-manifold, and let two knots in $N$ have orientation-homeomorphic exteriors that are not solid tori. The oriented knot complement conjecture. Ther…

Let $M$ be a knot complement, and let $\widehat{Z}(M)$ denote its homological block. Consider its positive or negative expansion, and let $\mathcal{P}_Q(\mathbf{x})$ denote a motiv…

$(a,t)$-deformed annihilation conjecture. The deformed series satisfies

Let $M_j=S^3\smallsetminus K_j$ be the complement of the twist knot $K_j$ with $j$ half-twists, and let $N$ be its thrice-punctured-sphere totally geodesic surface. For $j\geq 3$,…

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.