50 problems

Let $K$ be a hyperbolic knot in $S^3$. A realization assigns functions … for positive integers $N$, where $\mathsf{X}(K)$ parametrizes hyperbolic structures on the complement of…

Gordon's conjecture. If $S^3_{p/q}(K)$ is a Seifert fibered space, then

Chinburg–Reid–Stover's conjecture. In the notation above,

Fiberedness conjecture for hyperbolic knots yielding lens spaces. If a hyperbolic knot admits Dehn surgery yielding a lens space, then the knot is fibered.

Let $K\subset S^3$ be a hyperbolic knot, and suppose that a surgery on $K$ yields a lens space of order $r$. Bleiler--Litherland's conjecture. The order satisfies … The paper prove…

Let $K\subset S^3$ be a hyperbolic knot of genus $g$, and suppose that a surgery on $K$ yields a lens space of order $r$. Goda--Teragaito's conjecture. The order satisfies … The co…

Toroidal-slope bound conjecture. The knot $K$ has at most $3$ toroidal slopes.

Nonhyperbolic surgery conjecture. If $K(\gamma)$ fails to be hyperbolic, then $K(\gamma)$ is a Seifert fibered manifold or a toroidal manifold.

Optimistic-limit volume conjecture. There exists a solution $(x_0,y_0)$ to

Let $K(p,q,p-kq,1,1)$ be the knots of the second type in Theorem, with the parameters satisfying the hypotheses of that theorem. Hyperbolicity conjecture. All of the knots…

Let $L$ be a hyperbolic link, and let $J_N(L)$ be its colored Jones polynomial evaluated at $\exp\left(2\pi\sqrt{-1}/N\right)$. Write $\operatorname{vol}(L)$ for the hyperbolic vol…

Let $K$ be a hyperbolic knot with complement $M$, let $J_N(K,q)$ denote its $N$-colored Jones polynomial, and let $a$ be the ratio $N/k$. Consider the double-scaling limit … The pa…

Complexity equality conjecture. The inequality in the bounded-complexity result can be upgraded to equality for all 42 families of knots in this paper.

Linear growth conjecture. The entropy of braids in the sequence $\\{\beta^{mk+1}\\}_{m=M+1}^\infty$, a lower bound on the knot genus of…

Let $K$ be a hyperbolic knot in the 3-sphere $S^3$. A pair of Dehn surgeries on $K$ is chirally cosmetic if the resulting 3-manifolds are orientation-reversingly homeomorphic, and…

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

Generalized volume conjecture. For a hyperbolic knot $\mathscr{H}$, there exists a neighbourhood $U\in\mathbb{C}$ of $0$ such that, if $u\in U\setminus\pi\sqrt{-1}\mathbb{Q}$, then

Let $\mathscr{H}\subset S^3$ be a hyperbolic knot. Define its complex volume by … where $\operatorname{CS}^{\rm{SO(3)}}(\mathscr{H})$ is the Chern–Simons invariant modulo $\pi^2$.…

Fix a small cut-off value $d$ for the total rank of knot Floer homology. For $x$ a volume threshold, let $f(x)$ be the fraction of knots whose total knot Floer homology rank is les…

Let $K$ be a knot, let $\det(K)$ denote its determinant, let $\operatorname{Vol}(K)$ denote its hyperbolic volume, and let $c$ denote the crossing number. Asymptotic volume–determi…

Let $K$ be a knot, let $r(K)$ denote the total rank of its knot Floer homology, let $\operatorname{Vol}(K)$ denote its hyperbolic volume, and let $c$ denote the crossing number. As…

Let $G$ be a finite cyclic or dihedral group. A $G$-symmetry type of a prime knot is a symmetry type arising from an action of $G$ on the knot. Hyperbolic realization conjecture. F…

Let $\tau$ tend vertically to $0$, let $I^{\mathrm{rot}}(n,n')$ be the rotated 3D-index, and let $\sigma$ range over boundary-parabolic $\operatorname{PSL}_2(\mathbbm C)$-represent…

Let $K$ be a hyperbolic knot in $S^3$, and let $\widehat{\Phi}^{(\sigma_1)}_{K,n}$ and $\widehat{\Phi}^{(\sigma_2)}_{K,n}$ be the computable asymptotic series associated with the g…

For a representation $\rho:\pi\to {\rm SL}(2,\mathbb{C})$, let the twisted Alexander polynomial (TAP) associated to its conjugacy class be denoted by $\Delta_L^{[\rho]}$. If $K$ is…