93 problems

Radial limit conjecture. For each $b\in\operatorname{Spin}^c(M)/\{\pm1\}$, there exist topological invariants $\Delta_b\in\mathbb{Q}$, $c\in\mathbb{Z}_{\ge0}$, and…

Let $\overline{\mathfrak{g}}$ be a finite-dimensional simple Lie algebra, and let $\lambda\in \overline{P}_+\backslash\{0\}$ be a non-zero dominant integral weight. The relative as…

Reduced 2-loop identification conjecture.

First-polynomial identification conjecture.

Universal-invariant equivalence conjecture. The datum of the universal invariant $Z_\mathbb{D}(K)$ is equivalent to any of the three pieces of data in the preceding theorem.

Higher-order determination conjecture. The value of $Z_\mathbb{D}(K)$ is completely determined by the Alexander polynomial $\Delta_K$ and the family of polynomials $\rho_K^{i,0}$,…

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,…

Resolution conjecture. The generalized Kauffman bracket is given by

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.

Non-positive-curvature $C_2$ freeway basis conjecture. A collection of $C_2$ freeways of non-positive curvature, with one member in each switching class, constitutes a basis for

Non-positive-curvature $A_2$ freeway basis conjecture. The set of $A_2$ freeways of non-positive curvature having these points as endpoints is a basis for

Root-of-unity rationality conjecture. For every $k\ge 0$, one has $P_{K,\zeta,k}(t)\in\mathbb{Z}[\zeta ][t+t^{-1}]$. The source presents this as implied by the preceding cyclotomic…

Let $K$ be a knot and let $d\colon \mathbb{N}\to\{0,1,2,\ldots\}$ vanish except at finitely many elements. Define … Let $\Lambda$ be the cyclotomic completion containing the unifi…

Asymptotic ratio conjecture.

Let $p/q$ be a rational surgery coefficient, let $d$ be the inverse of $p$ modulo $q$, let $(a,b)\in\{0,1\}^2$, $n\in\mathbb{Z}/|q|\mathbb{Z}$, and $(\mu,\nu)\in\{\pm1\}^2$. Define…

Specialization conjecture.

Let $M$ be a weakly-gentle cusped manifold, and let $H_N(M)$ be its quantum dilogarithmic invariant. Let $\mathfrak c_{\mathcal I_{fc}}(M)$ denote its ${\mathcal I}_{fc}$-scissors…

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 geometrically to $M$, where $L_n$ consists of the simpl…

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 $K$ be a knot in $S^3$. Let $A(K)(L,M)$ denote its $A$-polynomial, and let $A_q(K)$ denote the $A_q$-polynomial of the recursion ideal of its colored Jones function. Under the…

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…

Center-trace equality conjecture.

Let $M$ be an oriented connected closed three-manifold, possibly equipped with a banded trivalent colored graph, and let $p$ be a prime. Define … For a connected three-manifold $M$…

For an integer $p$, define … Let $M_p$ be the closed three-manifold obtained from the three-sphere by $p$-surgery along the figure-eight knot. Stationary-phase conjecture for figur…

Volume conjecture for closed three-manifolds.