53 problems

Let $Y$ be an irreducible $L$-space, where an $L$-space is a rational homology sphere satisfying $\dim \widehat{\mathit{HF}}(Y)=|H_1(Y)|$. Ozsváth–Szabó's conjecture. The manifold…

Li–Ni's conjecture. Then $K$ is an iterated torus knot.

Let $K$ be a knot in $S^3$. An essential $2$–string tangle decomposition is one arising from a $2$–sphere intersecting $K$ transversely in four points such that the punctured spher…

Let $P$ be a pattern in the solid torus and let $K$ be a knot, with satellite knot $P(K)$. An L-space knot is a knot admitting a positive Dehn surgery yielding an L-space. Satellit…

Let $K$ be a nontrivial knot in $S^3$. The conjugate-slope property and the cofinal detection property are the properties defined in the paper. L-space knot characterization conjec…

Let $K$ be a nontrivial L-space knot in $S^3$, and let $g(K)$ be its Seifert genus. The conjugate-slope property at $n$ is the property defined in the paper for $n\in\mathbb N$. Co…

Let $W$ be an irreducible rational homology $3$-sphere that is not a lens space. Ba and Clay's conjecture. The following statements are equivalent: 1. The fundamental group of $W$…

Let $M$ be an orientable, compact, connected, irreducible $3$-manifold. Write $LO$ when the fundamental group of $M$ is left-orderable, $NLS$ when $M$ is not an $L$-space, and…

Let $Y$ be an irreducible rational homology $3$-sphere. A group is left-orderable if it admits a strict total order invariant under left multiplication, and a co-oriented taut foli…

Consider the filled manifolds obtained by Dehn filling orientable $1$-cusped hyperbolic $3$-manifolds in the censuses of at most $9$ tetrahedra, and let the manifolds for which the…

Let $K$ be the $(-2,3,2n+1)$-pretzel knot, where $n\geq 3$. Its genus is $g(K)=n+2$. A Dehn surgery slope is a rational number, and the resulting $3$-manifold is denoted by the sur…

Let $Y$ be an irreducible rational homology $3$-sphere. It is an L-space if its Heegaard Floer homology satisfies … A group is left-orderable if it admits a left-invariant total or…

Let $K$ be a knot or link and let $\Sigma(K)$ denote its double branched cover, equipped with a real structure $\iota$ and a compatible real $\mathrm{spin^c}$ structure…

Let $K\subset S^3$ be an L-space knot, meaning that some non-trivial positive Dehn surgery on $K$ is an L-space. A knot is strongly invertible if it admits an orientation-preservin…

Let $M$ be a hyperbolic L-space. A contact structure on $M$ is universally tight if its pullback to the universal cover of $M$ is tight. Min–Nonino's conjecture. No hyperbolic L-sp…

Symmetry-covering conjecture. These symmetries cover all symmetries of hyperbolic L-space knots.

Instanton L-space surgery conjecture. Under these assumptions, $S^3_{p/q}(K)$ must be an instanton L-space, meaning

Let $M$ be an irreducible rational homology $3$-sphere. The $L$-space conjecture. The following are equivalent: 1. $M$ carries a cooriented taut foliation. 2. $\pi_1(M)$ is left-or…

The $L$-space conjecture. The following statements are equivalent:

Let $Y$ be an irreducible rational homology 3-sphere. A non-L-space is a 3-manifold whose Heegaard Floer homology is not "small"; a group is left-orderable when it admits a total o…

Hyperbolic L-space conjecture. $M$ is an L-space if and only if it does not support a universally tight contact structure.

Let $Y$ be a 3-manifold. A left-orderable fundamental group is a fundamental group $\pi_1(Y)$ admitting a strict total order invariant under left multiplication. An L-space is a 3-…

Thin L-space knot conjecture. It is conjectured that the only thin, L-space knots are the torus knots $T(2,n)$.

Let $M$ be a compact, connected, orientable, irreducible, boundary-incompressible $3$-manifold with nonempty torus boundary, and let $D_{NLS}(M)$ and $D_{LO}(M)$ denote its sets of…

Let $M_1$ and $M_2$ be knot manifolds, and let $W=M_1\cup_f M_2$ be their union along their torus boundaries. For $*\in\{LO,CTF\}$, a rational slope is $*$-detected when it has the…