200 problems

Let $K$ be a knot. Its colored Jones polynomial is the sequence of quantum knot invariants indexed by colors, and its A-polynomial is the algebraic curve encoding the boundary char…

Let $K$ be a knot, let $G=SU(N)$, and consider the totally symmetric representation. Write … and let $\alpha_{-k-1}(K;N,q)$ denote the coefficients appearing in the inverted Habiro…

Let $\varphi$ be a pseudo-Anosov homeomorphism of a punctured surface, and let $M_\varphi$ be its hyperbolic mapping torus. The Bonahon–Wong–Yang invariants are defined at roots of…

Melvin–Morton–Rozansky conjecture. The semi-classical expansion is

Let $L(p,q)$ be a lens space, let $G$ be a simply connected compact simple Lie group with complexified Lie algebra $\mathfrak g$, and let $\Lambda^{R}$ be the coroot lattice. The C…

Let $K$ be a nice knot, let $F_K(x,q)$ be its $F_K$ series, let $N_p(K,x)$ be its ADO polynomial, let $\Delta_K(x)$ be its Alexander polynomial, and let $\zeta_p=e^{2\pi i/p}$ be a…

Let $Y$ be a closed oriented $3$-manifold. For each $\operatorname{SU}(2)$ Chern–Simons action $S\in\operatorname{CS}(Y)$, there exists a Puiseux series … such that the WRT invaria…

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 Chern–Simons theory be the three-dimensional quantum field theory associated with a gauge theory, and let a $3$-manifold be a smooth three-dimensional manifold. Witten's conjec…

Quantum A-polynomial annihilation conjecture. The series $F_K(x,q)$ is annihilated by the quantum $A$-polynomial:

Let $M_3$ be a 3-manifold and let ${\rm Tor}\,H_1(M_3;{\mathbb Z})$ denote the torsion subgroup of its first homology. A $q$-series $\hat{Z}_a(q)$ is associated to each…

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…

The Chern–Simons partition function is expressed in terms of quantum invariants of a link, with $P_{\overrightarrow{A}}(\mathcal{L};q,t)$ denoting the colored quantum invariant and…

Let $L$ be a link, let $LG^{n,m}$ denote the two-variable Links–Gould invariant derived from the super Hopf algebra $U_q\mathfrak{gl}(n|m)$, and let $\Delta_L$ be the Alexander–Con…

Let $\mathcal{H}^{\mathrm{HOMFLY}}(K)$ denote HOMFLY homology of a knot $K$, and let $\mathcal{H}^{\mathfrak{sl}(N)}(K)$ denote its $\mathfrak{sl}(N)$ homology. Let $d_N$ be the co…

Labastida–Mariño integrality conjecture. The reformulated invariants satisfy

Let $M$ be an integral homology three sphere. Let $+ default$ denote the generating series of the Ohtsuki invariants $default$ and let $default$ be the universal invariant of Le,…

Let $\lambda_n$ denote the $n$th Ohtsuki invariant of an integral homology 3-sphere. The source records the known facts $\lambda_1\in 6\mathbb Z$ and $\lambda_2\in 3\mathbb Z$. Lin…

Let ${\cal O}_k$ be the vector space of finite type invariants of $\mathbb{Z}$-homology 3-spheres of order at most $k$, and let $\lambda_n$ denote the coefficients in Ohtsuki's for…

Resolution conjecture. The generalized Kauffman bracket is given by

Tweenie-knot degree conjecture. The degree of $\Gamma_{2r}(\text{tweenie knot})$ is less than $r-1$.

Period $3p$ conjecture. $Z_p(RT)$ is a periodic map with period $3p$, for all $p$.

Let $Z^K$ denote the Kontsevich integral and $Z^{BT}$ the Bott–Taubes invariant. Kontsevich–Bott–Taubes conjecture. It is conjectured that … Both invariants are integral-formula co…

Domination conjecture. The perturbative invariant $\tau^{PSU(n)}(M)$ dominates the quantum invariants $\tau_r^{SU(n)}(M)$ for every positive integer $r$, not necessarily prime.

LMO's existence conjecture. Similar power series invariants should exist for the Lie group $SU(n)$ when $n>2$.