570 problems

Convexity question. Is the function $g$ convex?

A nontrivial group $G$ is left-orderable if it admits a total ordering invariant under left multiplication. An $L$-space is a rational homology $3$-sphere with minimal Heegaard Flo…

Let $\mathbb{K}$ be a subfield of $\mathbb{C}$, let $G$ be a countable group with a bound on the orders of its finite subgroups, and let $\mathrm{lcm}(G)$ denote the least common m…

Let $K\subset S^3$ be a lens space knot, meaning that some positive integer surgery on $K$ produces a lens space. Berge's construction gives a family of such knots. Berge conjectur…

Suppose that … is a tower of covers of congruence arithmetic hyperbolic 3-manifolds whose injectivity radius approaches infinity. Write $H_1(M_i,\mathbb{Z})_{tors}$ for the torsion…

Let $L=K_1\cup\dots\cup K_n$ be an $n$-component link in $S^3$. A 0-framed surgery on $L$ means surgery with framing zero on every component, and let $\#_n(S^2\times S^1)$ denote t…

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.

Let $K$ be a nontrivial knot in $S^3$. For surgery slopes $r$ and $s$, the manifolds $S^3_r(K)$ and $S^3_s(K)$ are purely cosmetic when they are homeomorphic as oriented manifolds.…

An R-link is a link $L$ such that 0-surgery on $L$ yields a connected sum of copies of $S^1\times S^2$. A 0-framed unlink is an unlink whose components all have framing zero, and a…

Let $M$ be a closed orientable irreducible atoroidal 3-manifold with positive first Betti number. An integral class is a class $a\in H^2(M;\mathbb{R})$ arising from integral cohomo…

L-space conjecture. The following statements are equivalent for $M$: (1) $M$ is a non-L-space. (2) $\pi_1(M)$ is left orderable. (3) $M$ admits a co-orientable taut foliation…

Let $f$ be a conservative partially hyperbolic diffeomorphism of a 3-manifold. Hertz–Hertz–Ures ergodic conjecture. If $f$ is non-ergodic, then there is a 2-torus tangential to…

Thurston's geometrization conjecture. Each component resulting from the geometric decomposition of $\mathcal{M}$ is geometric.

Let $M$ be a closed $3$-manifold. A pseudo-Anosov flow without negative lozenges is a pseudo-Anosov flow having no negative lozenges. Finiteness conjecture. There are finitely many…

Turaev–Viro invariants volume conjecture. For every such $M$,

Seifert-fibre-edge conjecture. Every one-vertex triangulation of a small Seifert fibre space has an edge isotopic to a Seifert fibre.

Let $Y$ be a closed, oriented three-manifold and $K$ a knot whose exterior is boundary-irreducible. A purely cosmetic surgery is a pair of distinct surgery slopes on $K$ producing…

Neuwirth conjecture. There exists a closed surface $F$ such that $F$ contains $K$ non-separatingly and is essential in the exterior of $K$.

Let $f$ be a partially hyperbolic diffeomorphism of a 3-manifold. Pujals' classification conjecture. If $f$ is transitive, then $f$ is finitely covered by one of the following: a p…

Let $Y$ be an irreducible homology 3-sphere and suppose that $Y$ is an L-space. Ozsváth–Szabó's conjecture. Then $Y$ is homeomorphic to either $S^3$ or the Poincaré homology sphere…

Let $M$ be a hyperbolic $3$-manifold, and let $\pi_1(M)$ be its fundamental group. Lubotzky–Sarnak conjecture. The group $\pi_1(M)$ does not have property $(\tau)$. This is present…

Let $(M,\gamma)$ be a taut sutured manifold, meaning a sutured manifold satisfying the tautness condition, and suppose that $\chi(M)<0$. A finite-sheeted cover is a covering … with…

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 $M=M_1\# M_2$, where each $M_i$ is a 3-manifold, possibly with holes, and neither $M_i$ is equal to $S^3$. Let ${\cal S}_{2,\infty}(M)$ denote the Kauffman bracket skein module…

Let $F$ be a closed surface, let $x_i\in F\times\{0\}$ for each $i$, and let ${\cal S}_{2,\infty}(F\times I,\{x_i\}_1^{2n};R,A)$ denote the relative Kauffman bracket skein module w…