116 problems

For every closed orientable hyperbolic $3$-manifold $M$, the odd-level $SO(3)$ Turaev–Viro invariants satisfy …

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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 $\Sigma_{g,n}$ be a compact oriented surface, and let … be its mapping class group, with homeomorphisms and isotopies fixed on the boundary. For an odd integer $r\geq 3$, let…

For an oriented compact three-manifold $X$, let $G$ be a semisimple, simply connected Lie group, and let $\mathcal{Z}_G^{(K)}(X)$ denote the Reshetikhin–Turaev TQFT invariant at le…

AJ conjecture. For every knot $K$ in $S^3$, $\varepsilon(\alpha_K)$ is equal to $A_K(M,L)$, up to multiplication by a polynomial depending only on $M$.

Let $K\subset{\mathbb S}^3$ be a hyperbolic knot, let $\gamma$ be a simple closed curve on $\partial\nu(K)$, and let $\rho_0$ be the discrete faithful representation associated wit…

Gukov–Manolescu conjecture. The series satisfies

Let $M$ be a hyperbolic $3$-manifold with a regular triangulation $\mathcal{T}$ and NeumannZagier datum, including a choice of quad type with $\det \mathbf{B}\neq 0$. Let…

Gukov–Pei–Putrov–Vafa conjecture. In the above situation,

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…

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…

The volume conjecture for polyhedra. Then

Cyclotomic expansion conjecture. For any knot $\mathcal{K}$, there exist Laurent polynomials $H_k^{(n)}(\mathcal{K})\in\mathbb{Z}[q,q^{-1}]$, independent of $N$, such that

Let $K\subseteq S^3$ be a knot. Write $F_K(x,q)$ for the knot invariant, set $q={\rm e}^{\hbar}$, and keep $x=e^{n\hbar}$ fixed, where $n$ is the color of $K$. Let $\Delta_K(x)$ be…

Gukov's conjecture. The function $\widehat{Z}_{\boldsymbol{a}}(q)$ is a depth $r$ quantum modular form whose quantum set is a subset of $\mathbb Q$. Moreover, for any unimodular pl…

Let $\mathcal{H}_g^{V(r)}$ be the small Heisenberg homology, let $\mathcal{H}_g^{V,\dagger}$ be the direct sum of the standard twisted homology groups, and let…

Let $K$ be a knot. Denote by $A_K(l,\alpha)$ its $A$-polynomial, by $A_q(K)(l,\alpha^2)$ the specialization at $q=1$ of the $A_q$-polynomial, and let $\varepsilon$ be the evaluatio…

For every knot $K\subset S^3$, let $F_K^N(x,q)=F_K^{\mathfrak{sl}_N,\mathrm{sym}}(x,q)$ denote the series associated with symmetric representations, let $\Delta_K(x)$ be the Alexan…

Ohtsuki's factorial integrality conjecture. For every $n\geq 1$ and every integral homology 3-sphere $M$,

mod $p$ non-vanishing conjecture. The function $\widetilde{\tau }_p(M)$ is nowhere zero on $\mathbb{F}_p^\times$. The source says this is supported by computer calculations and giv…

$p$-adic non-vanishing conjecture. The function $\tau ^p(M)$ is nowhere zero on $U(\mathcal{O}_p)$. The source presents this as a generalization of the root-of-unity non-vanishing…

Let $M$ be an integral homology sphere, let $\zeta _8$ be a primitive eighth root of unity, and define $\widetilde{\tau }_8(M)=\sqrt{-1}^{\lambda(M)}\tau _8(M)$. Eighth-invariant n…

Let $M$ be an integral homology sphere and let $\tau _\zeta(M)$ denote its Witten–Reshetikhin–Turaev invariant at a root of unity $\zeta$. Non-vanishing conjecture. For every root…

Let $X$ be a Seifert manifold with base $S^2$, and let $(l,\underline{n}')\in\mathcal I_2^b$. Suppose that there do not exist elements $g_1,\ldots,g_n\in\operatorname{SU}(2)$ satis…