24 problems

Let $M$ be a finite-volume hyperbolic 3-manifold, and let $TV_r(M;e^{\pi i/r})$ be its Turaev–Viro invariant for odd integers $r\geq 3$. The Chen–Yang volume conjecture. The growth…

Let $M$ be a hyperbolic $3$-manifold, either closed, cusped or with totally geodesic boundary, and let $\operatorname{Vol}(M)$ denote its volume. As $r$ varies over the odd integer…

Let $M$ be a compact orientable 3-manifold with empty or toroidal boundary. For odd integers $r\geq 3$, let $TV_r(M;e^{\pi i/r})$ denote the Turaev–Viro invariant at the specified…

Volume conjecture. As $r$ ranges over all odd integers and $q=e^{\frac{2\pi i}{r}}$,

Let $L(7,1)$ and $L(7,2)$ be the indicated lens spaces, and suppose there exists a generalized $E_6$-subfactor with group symmetry $G=\mathbb{Z}/7\mathbb{Z}$. Let the Turaev–Viro–O…

Let $M$ be a cusped hyperbolic $3$-manifold, let $TV(M,q)$ denote its Turaev–Viro invariant at a root of unity $q$, and let $\operatorname{Vol}(M)$ denote its hyperbolic volume. Pa…

Let $M$ be a compact orientable 3-manifold with empty or toroidal boundary. Define … where $r$ runs over all odd integers, and let $\operatorname{Vol}(M)$ be the simplicial volume,…

Let $\mathcal{L}$ be a hyperbolic link in $S^3$, and let $J_N(\mathcal{L};e^{\frac{2\pi\sqrt{-1}}{N+\frac{1}{2}}})$ denote its colored Jones polynomial evaluated at the shifted roo…

Let $M$ be a compact, oriented $3$-manifold with empty or toroidal boundary, and let … The manifold $M$ is $q$-hyperbolic when $\textit{lTV}(M)>0$, and $\lVert M\rVert$ denotes its…

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…

Detcherry–Kalfagianni simplicial-volume conjecture.

Let $M$ be a compact and orientable $3$-manifold with empty or toroidal boundary. For odd integers $r$, set … Let $v_3\approx 1.0149$ be the volume of a regular ideal tetrahedron,…

Let $L$ be a link in $\mathbb S^3$, and let $TV_r(\mathbb S^3\backslash L,t)$ denote the Turaev–Viro invariant of its complement at level $r$. Let $v_3$ be the volume of a regular…

Detcherry–Kalfagianni–Yang generalized Turaev–Viro volume conjecture.

Unified potential-function conjecture. The function recovers the parametrized potentials, $\Phi(M/(N+1/2),z)=\Phi_M^{(s)}(z)$ and $\Phi(1,z)=\Phi(z)$; it admits a smooth choice of…

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

Let $\Sigma$ be a compact oriented surface and let $f\in\mathrm{Mod}(\Sigma)$ be pseudo-Anosov. Let … be the mapping torus of $f$, and let $TV_r(M_f)$ denote its $SO(3)$ Turaev–Vir…

Let $L\subset S^3$ be a link, and let $||L||$ denote its simplicial volume, defined as the sum of the volumes of the hyperbolic pieces in the JSJ decomposition of…

Let $T_{(m,n)}$ be the $(m,n)$-torus link in $S^3$, with $m<n$. Let $TV_r(S^3\setminus T_{(m,n)})$ be its Turaev–Viro invariant. Chen–Yang integrality conjecture. If $r$ is coprime…

Let $K$ be a knot in $S^3$, and let $TV_r(K)$ denote the corresponding Turaev–Viro invariant, evaluated at the allowed roots of unity $q$. Chen–Yang unknot detection conjecture. ……

For a closed 3-manifold $M$, let ${\mathcal X}^{TV}(M)$ denote the set of homeomorphism classes of closed orientable 3-manifolds having the same Turaev–Viro invariants as $M$. Let…

Let $M$ and $N$ be closed irreducible geometric 3-manifolds. Their abelian and $SU(2)$ Turaev–Viro invariants are the collections of invariants associated to abelian and $SU(2)$ th…

Let $\mathcal{C}$ be a spherical category, not necessarily modular, and let $Z(\mathcal{C})$ denote its Drinfeld center. For a 3-manifold $M$, write $Z_{TV,\mathcal{C}}(M)$ for the…

Let $\mathcal{C}$ be a spherical fusion category over an algebraically closed field $\Bbbk$ with invertible dimension, and let $\mathcal{Z}(\mathcal{C})$ denote its center. For a…