39 problems

Let $M$ be a rational homology $3$-sphere. Its fundamental group is left-orderable if it admits a left-invariant total order, and $M$ supports a co-orientable taut foliation if it…

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 knot in $S^3$, and let $g(K)$ denote its Seifert genus. The conjugate-slope property at $n$ is the property defined in the paper for $n\in\mathbb N$, and Ni…

L-space conjecture. The following statements are equivalent:

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…

L-space conjecture. $M$ admits a coorientable taut foliation if and only if $\pi_1(M)$ is left-orderable, and this happens if and only if $M$ is not an $L$-space.

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,…

Non-left-orderability conjecture. No irreducible lattice in $G$ has a left-invariant total order. Equivalently, no such lattice has a nontrivial, orientation-preserving action on t…

L-space conjecture. These three properties are equivalent.

Let $M$ be a closed, connected $3$-manifold. An L-space is a rational homology $3$-sphere $M$ satisfying … A group is left-orderable if it is nontrivial and admits a total order in…

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…

Hyperbolic-knot L-space conjecture. Either $K(r)$ is left-orderable for every $r\in(-\infty,5)$, or $K(r)$ is left-orderable for every $r\in(-5,\infty)$.

Small-slope L-space conjecture. $K(1/q)$ is left-orderable unless $K=T(2,3)$ and $q=1$, or $K=T(2,-3)$ and $q=-1$.

L-space conjecture for winding-one satellites. Every rational surgery $K(r)$ is left-orderable.

L-space conjecture. The following are equivalent:

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…

The L-space conjecture. The following are equivalent:

L-space conjecture. The manifold $Y$ is not an L-space if and only if $G$ is left-orderable.

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…

Let $M$ be a closed orientable irreducible $3$-manifold with a left-orderable fundamental group, and let $M_0=M-\operatorname{Int}(B^3)$. Suppose that $\mathcal{F}$ is a Reebless c…

Let $K$ be a knot in $\mathbb{S}^3$, and let $g(K)$ denote its genus. L-space surgery conjecture. If $\pi_1(\mathbb{S}^3_{p/q}(K))$ is non-left-orderable for some $p/q>0$, then…

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…