200 problems

Let $\mathfrak{S}$ be a triangulable pb surface and let $\lambda$ be an ideal triangulation. Let $\overline{\mathscr S}_\omega(\mathfrak{S})$ be the reduced stated ${\rm SL}_n$-ske…

Let $K$ be a hyperbolic knot, meaning that its complement admits the hyperbolic Thurston geometry, and let $J_r(K;q)$ denote its $r$-th colored Jones polynomial. Kashaev–Murakami–M…

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…

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

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…

Let $K$ be a hyperbolic knot, let $J_r(K;q)$ be its $r$-th colored Jones polynomial, and let $A_*$ denote the geometric flat connection associated with the canonical hyperbolic str…

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…

Let $\mathcal L$ be an $r$-component link, with components colored by representations $R_1,\ldots,R_r$, and let $f_{(R_1,R_2,\ldots,R_r)}(q,\lambda)$ denote its reformulated invari…

Let $M$ be a closed and oriented $3$-manifold. For each positive integer $k$, let $Z_k(M)\in\mathbb{C}$ denote the $\operatorname{SU}(2)$ WRT invariant for $M$ at level $k$. 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…

Let $M$ be an integral homology sphere, and let $\tau ^O(M)=1+\sum_{n=1}^\infty \lambda _n(M)(q-1)^n\in \mathbb{Q}[[q-1]]$ be its Ohtsuki series. Lawrence's conjecture. The series…

Roger–Yang representation-theoretic conjecture. The representation theory of the Roger–Yang skein algebra should encode hyperbolic geometric information about the punctured surface…

Let $\beta$ be a braid and let $L$ be its closure. Let ${\large {\Huge {\Gamma^R}}}(L)\in\mathbb L^R$ denote the refined universal ADO intersection, and let…

In quantum Chern–Simons theory, the Kashaev invariant of a knot and the WRT invariant of a $3$-manifold are examples of quantum invariants with asymptotic expansions at roots of un…

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…

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…

Let $T$ be a spherical tetrahedron whose edge lengths are $l_{ij}=\pi\alpha_{ij}$ for $1\leq i<j\leq4$, with associated dihedral angles $\theta_{ij}$, volume $V$, and spherical Gra…

Let $b$ be an odd positive integer, let $cmathcal M_b$ be the class of 3-manifolds under consideration, and let $cmathcal Gamma_b$ be the subring of unified invariants whose evalua…

Let $\mathcal C$ be a modular tensor category, and let $\rho_{\mathcal C}^{\lambda,1}$ be its modular representation; equivalently, consider the corresponding representation for a…

The original Kontsevich integral is a universal finite-type invariant of 3-manifolds, and $h_\infty$ denotes the Chern–Simons invariant constructed earlier in the paper. Equality c…

Let $F8$ denote the figure-eight knot, viewed as a genus-one fibered knot, and let $Z_p(F8)$ be the map induced by the monodromy of its zero-framed surgery, acting on the correspon…

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

Let $H$ be a finite-dimensional Hopf algebra. Kuperberg's invariant assigns an invariant of framed three-manifolds to $H$. The quantum double of $H$ is quasitriangular but need not…

Let $H$ be a finite-dimensional Hopf algebra. Its quantum double is factorizable and unimodular, although it is often not ribbon, and suppose an invariant of framed three-manifolds…