Twist families of L-space knots, their genera, and Seifert surgeries

Abstract

Conjecturally, there are only finitely many Heegaard Floer L-space knots in $S^3$ of a given genus. We examine this conjecture for twist families of knots $\{K_n\}$ obtained by twisting a knot $K$ in $S^3$ along an unknot $c$ in terms of the linking number $\omega$ between $K$ and $c$. We establish the conjecture in case of $|\omega |\ne 1$, prove that $\{K_n\}$ contains at most three L-space knots if $\omega =0$, and address the case where $|\omega |=1$ under an additional hypothesis about Seifert surgeries. To that end, we characterize a twisting circle $c$ for which $\{(K_n,r_n)\}$ contains at least ten Seifert surgeries. We also pose a few questions about the nature of twist families of L-space knots, their expressions as closures of positive (or negative) braids, and their wrapping about the twisting circle.

In addition to addressing typos, V2 restructures, expands, and corrects parts of V1 and incorporates comments of referees. Accepted by Comm. Anal. Geom

References in corpus (2)

• Floer homology and knot complements
• Surgeries, sharp 4-manifolds and the Alexander polynomial

Cited by in corpus (3)

• Seifert vs slice genera of knots in twist families and a characterization of braid axes
• Satellite knots and L-space surgeries
• On L-space knots obtained from unknotting arcs in alternating diagrams