48 problems

Limsup volume conjecture. For every non-splittable link $L$,

Parameterized Volume Conjecture. There exists an open subset $\mathcal{O}_K\subset\mathbb{C}$ such that for every $u\in\mathcal{O}_K$,

For a knot $K$ in $S^3$, define the sequence … where $J_K(n)$ is the colored Jones polynomial. The eventual monotonicity conjecture. For every knot $K$ in $S^3$, the sequence $a_n$…

Let $M$ be a cusped manifold, define … and let $H_N(M)$ denote the proposed quantum hyperbolic invariant. Here ${\rm CS}(M)$ and ${\rm Vol}(M)$ are respectively the metric Chern–Si…

Let $M$ be a cusped manifold, and let $(W_n,L_n,\rho_n)$ be a sequence of compact hyperbolic Dehn fillings of $M$ converging to $M$, with $L_n$ the link of short simple geodesics f…

Real Volume Conjecture. For any $(W,L,\rho)$,

Volume conjecture for closed three-manifolds.

Turaev–Viro invariants volume conjecture. For every such $M$,

Let $K$ be a hyperbolic knot or link, so that $S^3\setminus K$ admits a hyperbolic structure. Let $J_{N-1}(K)$ be the normalized colored Jones polynomial at $q=\exp(\pi i/N)$, and…

Let $K$ be a framed knot or link in $S^3$. Let $V_N(K)$ be the colored Jones polynomial corresponding to the $(N+1)$-dimensional irreducible representation of the quantum group…

Let $M$ be a hyperbolic $3$-manifold, either closed, with cusps, or compact with totally geodesic boundary. Chen–Yang's Turaev–Viro volume conjecture. As $r$ varies along the odd n…

Let $K$ be a hyperbolic knot, and call a diagram of $K$ arc-faithful when its associated octahedral coloring has the required non-pinched property and holonomy equal to the complet…

Let $K\subset S^3$ be a knot, and let $\|K\|$ denote its simplicial volume, known here to be the sum of the hyperbolic volumes of the hyperbolic pieces in the knot complement. Let…

The volume conjecture for polyhedra. At $q=e^{\frac{2\pi\sqrt{-1}}{r}}$, as $r$ runs over all the odd integers,

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. Let $\operatorname{cv}(\mathscr{H})$ be its complex volume, and let $T(\mathscr{H})$ be the adjoint cohomological Reidemeister to…

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 $\pi^2$.…

Let $K$ be a knot in $S^3$. The geometric quantized space for the $SO(3)$-Witten–Chern–Simons theory of the torus is the alternating subspace … of holomorphic sections of…

Let $S$ be an oriented surface with negative Euler characteristic, let $\varphi$ be a periodic diffeomorphism, and let $[\gamma]$ be a $\varphi$-invariant smooth character with…

The volume conjecture for polyhedra. Then

Let $P$ be a hyperbolic polyhedron with dihedral angles $\alpha_1,\dots,\alpha_m$ at edges $e_1,\dots,e_m$, and let $\Gamma$ be its $1$-skeleton. Let $col_r$ be a sequence of $r$-a…

Let $K$ be a knot, let $N$ be a strictly positive integer, let $n$ be an integer parameter, and let $\zeta$ be a primitive $N$-th root of unity. Write $\langle K\rangle_{N,n}$ for…

Toroidal simplicial-volume conjecture.

Toroidal volume conjecture.