20 problems

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…

L-space conjecture. The following statements are equivalent:

Let $M$ be a closed 3-manifold. An almost periodic action of a finitely generated group $G$ on $\mathbb R$ is a faithful action in which the generators act uniformly bilipschitzly,…

Let $L$ be a quasi-alternating link, and let $IQ(L)$ denote its involutory quandle. A quandle is left-orderable if it admits a strict total order invariant under all left translati…

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 $M$ be a knot manifold. Write $\mathcal{S}(M)$ for the set of slopes on $\partial M$, $\mathcal{D}_{ord}^{wk}(M)$ for the set of weakly order-detected slopes, and $LO(M)$ for t…

Let $M_1$ and $M_2$ be knot manifolds, and let … where $f:\partial M_1\xrightarrow{\;\cong\;}\partial M_2$ is a homeomorphism. A slope $[\alpha_i]$ belongs to the set…

Let $M$ be a knot manifold. The sets of regular order-detected and weakly order-detected slopes on its boundary are denoted by $4\mathcal{D}_{ord}(M)4$ and…

Let $M$ be a closed orientable irreducible $3$-manifold such that $\pi_1(M)$ is left orderable, and let $M_0=M-\operatorname{Int}(B^3)$. Let $\mathcal{F}$ be a foliation of $M_0$ c…

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 $M$ be an irreducible, rational homology $3$-sphere other than $\mathbb{S}^3$. The L-space conjecture. The following are equivalent: 1. The fundamental group of $M$ is left-ord…

The $L$-space conjecture. The fundamental group $\pi_1(M)$ is non-left-orderable if and only if $M$ is an $L$-space.

Let $K$ be an L-space knot, and let $\mu$ and $\lambda$ represent a meridian and a longitude in the knot group. For a homomorphism $\rho$ from $\pi_1(S^3\setminus K)$ to…

Let $M$ be a rational homology $3$-sphere, meaning that $H_1(M;\mathbf{Q})=0$. Its Heegaard Floer homology has rank at least $\left|H_1(M;\mathbf{Z})\right|$, and $M$ is an L-space…

Let $M$ be a hyperbolic $\mathbb{Z}$-homology solid torus, and let $M(0)$ denote its longitudinal filling. Assume that $M(0)$ is hyperbolic and that its holonomy representation has…

Let $s\ge 3$ be an integer, and let $M_{p/q}$ be the manifold obtained by Dehn surgery with slope $\frac{p}{q}$ along the $(-2,3,2s+1)$-pretzel knot. Pretzel-knot surgery conjectur…

L-space conjecture. The following statements are equivalent:

Interval-shape conjecture. The set $\mathcal{S}_{LO}(K)$ coincides with one of

Boyer–Gordon–Watson's surgery dichotomy conjecture. If $K$ is not a cable of a nontrivial knot, then

Let $Y$ be a closed, connected $3$-manifold. A group is left-orderable if it is nontrivial and admits a total order invariant under left multiplication, and $Y$ is an L-space if it…