27 problems

Let $n$ be a positive integer, and consider all prime knots with $n$ or fewer crossings. The genericity conjecture for hyperbolic knots. The percentage of these knots that are hype…

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…

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…

Asymptotic expansion conjecture. As $N\to\infty$ and $k\to\infty$ with $u$ fixed,

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…

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 $M$ be an oriented compact closed $3$-manifold, and let $K$ be a hyperbolic knot in $M$. For a fully balanced shaped ideal triangulation $X$ of the complementary space of $K$ i…

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…

Slope-signature deviation conjecture. There are constants $b$ and $c$ such that, for any hyperbolic knot $K$ in $S^3$,

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

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

Integrality conjecture. Any Seifert-fibered surgery on a hyperbolic knot in $S^3$ is integral.

Complete-structure dominance conjecture. The colored Jones terms with $s\in U$ dominate those with $s\notin U$:

Parametrized asymptotic expansion conjecture. The family determines the asymptotic expansion

Let $K$ be a hyperbolic knot, let $g(K)$ denote its genus, and let $\rho_{\rm holo}:\pi(K)\to\operatorname{SL}(2,\mathbb{C})$ be a lift of the discrete and faithful representation…

Let a knot have weak property PT when it satisfies the property defined in the paper. Positive-density weak property PT conjecture. There exist $\varepsilon>0$ and $N>0$ such that,…

Let $K\subset S^3$ be a hyperbolic oriented knot, and let $\pi_K$ denote its knot group. Let $\alpha\colon \pi_K\to \operatorname{SL}(2,\mathbb{C})$ be a lift of the discrete and f…

Let $K$ be a hyperbolic knot, and let $x$ be an algebraic solution of the periodicity equations associated with the braid-induced triangulation of $S^3\setminus K$. For the $j$th o…