25 problems

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 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 $W$…

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

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

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

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…

Let $M$ be an irreducible, rational homology $3$-sphere that is not a lens space. Circular orderability version of the L-space conjecture. The following are equivalent: 1. The fund…

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 $L$ be a prime, fibred, strongly quasipositive link. Write $k(L)$ for its BKL-exponent, defined as the maximum integer $k$ such that $L$ is the closure of $\delta_n^kP$ for som…

Let $L$ be a link in $S^3$, and for each integer $n\geq 2$ let $\Sigma_n(L)$ denote the canonical $n$-fold cyclic cover branched over $L$. A link is simply laced arborescent if it…

Let $K$ be a knot in $S^3$. An essential tangle decomposition is an essential $n$–string tangle decomposition for some integer $n$, meaning that there is a $2$–sphere intersecting…

Let $Y$ be an L-space, and let $K\subset Y$ be a Floer simple knot with irreducible complement. Write $[K]^{\perp}$ for the orthogonal complement of the homology class $[K]$ under…

Let $K_1$ and $K_2$ be knots in $S^3$, and let $M_i=S^3\smallsetminus\nu(K_i)$ for $i=1,2$. Let $h$ identify the torus boundaries of $M_1$ and $M_2$, and let…

Alternating-surgery tangle-replacement conjecture. Every alternating surgery arises as tangle replacement on an almost-alternating diagram of the unknot; equivalently,

Splicing conjecture. The manifold $Y$ is an $L$-space if and only if all of the following conditions hold:

Classification conjecture. The set of all $L$-space two-bridge links is

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…

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

Let $Y$ be an irreducible homology sphere, and suppose that it contains an incompressible torus, meaning an embedded torus whose inclusion induces an injective map on fundamental g…

A rational homology 3-sphere $Y$ is an L-space if the rank of its Heegaard Floer homology group $\widehat{\operatorname{HF}}(Y)$ equals the order of $H_1(Y;\mathbb{Z})$. An irreduc…