27 problems

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, and let $g(K)$ denote the minimal genus of a Seifert surface for $K$. Lidman–Sivek conjecture. For every $L$-space knot $K$, … This conjecture would i…

Thin surgery discrepancy question. Does there exist a strongly invertible L-space knot $(K,\Phi)$ with a thin surgery such that $\varkappa(K,\Phi)$ is not thin?

Let $K$ be a hyperbolic L-space knot, let $b(K)$ denote its bridge number, and suppose that $b(K)=b$. Let $\Delta_K(t)$ denote the Alexander polynomial of $K$. KriMor's conjecture.…

Arbitrarily wide varkappa-invariant question. Does there exist a strongly invertible L-space knot $(K,\Phi)$ such that $\varkappa(K,\Phi)$ has width at least $n$?

Thin-slope formal semi-group question. Is every sufficiently large slope of $K$ a thin slope if and only if the formal semi-group of $K$ is not a semi-group?

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 periodic or freely periodic L-space knot. Iterated torus knot conjecture. $K$ is an iterated torus knot. The paper presents this as a conjecture motivated by computati…

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…

Let $\mathcal{C}$ denote the smooth knot concordance group. An L-space knot is a knot admitting a positive Dehn surgery yielding an L-space. Abe–Tagami's conjecture. The set of L-s…

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

Let $K$ be a positive 1-bridge braid in $S^3$, meaning a knot isotopic to a union of a positive braid on the boundary of a solid torus and a bridge in a meridional disk. Rasmussen'…

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, and let $U$ denote the unknot. Lidman–Vafaee's converse conjecture. Under conditions that the pattern is embedded “nicel…

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…

A closed braid in the solid torus satisfying the criterion in the source is called an L-space braid. A positive one-bridge braid is a one-bridge braid with the positivity condition…

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…

Let $\mathcal{A}$ be the subgroup of the smooth concordance group generated by algebraic knots, and let $\mathcal{L}$ be the subgroup generated by $L$-space knots. A subgroup is fr…

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 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…