570 problems

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p$. Consider $N_\Phi$ a Seifert fibered space with base orbifold $S^2(2,2,2,3)$ and gluing matrix $\Phi$ $=$ ……

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 $M$ be an oriented compact $3$-manifold with a torus boundary, and let $\mathrm{R}^{\rm para}_{\mathrm{SL}_2(\mathbb{C})}(M)$ be the set of conjugacy classes of boundary-parabo…

Let $M$ be an oriented closed $3$-manifold, let $G$ be a simply connected complex simple Lie group, and let $\mathrm{R}^{\rm irr}_G(M)$ be the character variety of irreducible repr…

Cabling Conjecture. If $K$ is a knot in $S^3$ which has a reducible surgery, then $K$ is a cabled knot and the reducing slope is given by the cabling annulus.

Regularized $+p$-surgery formula. When the $+p$-surgery formula of Gukov and Manolescu converges, it should be used; when it does not converge, the displayed identity should regula…

Strong invertibility conjecture. Every L-space knot in $S^3$ is strongly invertible: there is an orientation-preserving involution of $S^3$ whose fixed-point set is a circle inters…

Let $K$ be a hyperbolic knot in $S^3$, and let $b[?25l\cT_Kb[?25h \in \C[t^{\pm1}]$ be its hyperbolic torsion polynomial. The Seifert genus of $K$ is denoted by $g(K)$. Hyperbolic…

Let $M$ be a $3$-manifold, let $n$ be a positive integer, and let $\rho$ be a boundary-unipotent representation of $\pi_1(M)$ in $\mathrm{SL}(n,\mathbb C)$ or…

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 $M$ be a simply connected $3$-manifold, possibly with holes, and let $S_{2,\infty}(M)$ denote its Kauffman bracket skein module at parameter $2$ and infinity. Traczyk's conject…

Let $K_1,K_2\subset S^3$, write $K=K_1\#K_2$, and let $E(K_i)$ denote the exterior of $K_i$. Let $t(K_i)$ denote the tunnel number, and call a Heegaard splitting minimal genus if i…

Let $K_1$ and $K_2$ be knots in $S^3$, and write $K=K_1\#K_2$. Let $E(K)$ denote the exterior of $K$, and let $t(K)$ denote its tunnel number. A Heegaard splitting is weakly reduci…

McMullen's conjecture. The convex core $C(N)$ does not contain an embedded ball of radius $L$, where $L$ depends on the number of generators of $\pi_1(N)$.

Let $M$ be a hyperbolic manifold containing an incompressible surface $S$. For a manifold $N$ with acylindrical boundary, let $Vol(N)$ denote the volume of the components of $N$ th…

Virtual positive first Betti number conjecture. $M$ has a finite cover $ctilde{M}$ with

Let $M$ be a closed, irreducible $3$-manifold and let $\mathcal F$ be a maximal embedded collection of orientable, incompressible surfaces. If $M-N(\mathcal F)$ has $n$ components,…

Let $G$ be a finitely generated group with $g$ generators. For a hyperbolic $3$-manifold $N$, write $\operatorname{inj}_N(x)$ for the injectivity radius at $x$, and let the convex…

Let $K$ be a null-homologous oriented $k$-component link with zero pairwise linking numbers in a rational homology sphere $\Sigma$, and let $L$ be an admissible link of $ell$ compon…

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.

Let $M^{3}$ be a closed, prime, atoroidal $3$-manifold. A locally homogeneous Riemannian metric is a Riemannian metric such that for any two points $p$ and $q$ there is an isometry…

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subg…

Let $N$ be a compact, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold whose boundary is a non-empty collection of incompressible tori and Klein bottles. Assume that every subg…

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. Rational subgroup conjecture. Every subgroup of $\pi_1(M)$ which embeds i…

Let $M$ be a closed, connected, ${\mathbf{P}^2}$-irreducible $3$-manifold with infinite fundamental group. For a subgroup $H\leq\pi_1(M)$, write … for its normalizer. Normalizer Co…