17 problems

Watson's conjecture. $K$ is an L-space knot if and only if $[?]$ is supported in a single diagonal grading

Let $K$ be a periodic L-space knot with quotient knot $\overline{K}$. Quotient L-space knot conjecture. $\overline{K}$ is an L-space knot. The claim concerns how the L-space-knot p…

Let $K$ be a periodic or freely periodic L-space knot, and let $\Delta_K(t)$ denote its Alexander polynomial. Cyclotomic Alexander polynomial conjecture. $\Delta_K(t)$ is a product…

Let $K$ be a hyperbolic L-space knot, let $g$ be its Seifert genus, and let $n$ be its braid index. Let $\Delta_K(t)$ denote its Alexander polynomial. The hyperbolic L-space knot Al…

Let $K$ be an L-space knot, meaning a knot admitting a Dehn surgery yielding an L-space. Let $b(K)$ denote its bridge index and $\beta(K)$ its braid index. The L-space knot index c…

Azumaya-positive knot conjecture. $K$ is not an $L$-space knot.

Let $P$ be a pattern in the solid torus, let $K$ be a knot of genus $g$, let $U$ denote the unknot, and write $P(U,-n)$ for the pattern knot after $-n$ full twists. Satellite L-spa…

Let $P$ be a pattern in the solid torus, let $K$ be a knot, let $U$ denote the unknot, and write $P(K)$ and $P(U)$ for the corresponding satellite knots. An L-space knot is a knot…

Let $\{K_n\}$ be a twist family of knots, obtained by twisting along a fixed unknot disjoint from the initial knot, and let $g(K_n)$ denote the genus of $K_n$. Twist-family finiten…

Let $N\geq 0$ be an integer, and let $g(K)$ denote the genus of a knot $K$. Finiteness conjecture for bounded genus. There are only finitely many L-space knots $K$ with…

Let $K$ be an L-space knot with knot Floer homology $\widehat{\operatorname{HFK}}(K)$. Hedden–Watson's finiteness conjecture. There are only finitely many other knots whose Floer h…

L-space-knot linear-independence conjecture. The set of $L$-space knots in $S^3$ is linearly independent in the knot concordance group. In particular, if $L$-space knots $K_0$ and…

Let $K$ be a knot whose Floer homology agrees with that of an L-space knot, and suppose the Alexander polynomial of that L-space knot has all its roots on the unit circle. Li–Ni co…

Let $K$ be an L-space knot, meaning a knot for which $S^3_n(K)$ is an L-space for some $n>0$. An essential Conway sphere is an essential $4$-punctured sphere in the knot complement…

Let $K$ be an L-space knot, meaning a knot for which $S^3_n(K)$ is an L-space for some $n>0$. A strong inversion is an involution of the ambient $3$-manifold preserving the knot wh…

Let $K$ be an L-space knot, meaning a knot for which $S^3_n(K)$ is an L-space for some $n>0$. Let $\widehat{HFK}(K)$ denote its hat knot Floer homology. L-space-knot finiteness con…

Recall that an L-space is a rational homology sphere $Y$ satisfying … and a knot in $S^3$ admitting an L-space surgery is called an L-space knot. A strong inversion on a knot is an…