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

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 $\lbrace K_n \rbrace$ 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 the case of $\lvert \omega \rvert \neq 1$, prove that $\lbrace K_n \rbrace$ contains at most three L-space knots if $\omega = 0$, and address the case where $\lvert \omega \rvert = 1$ under an additional hypothesis about Seifert surgeries. To that end, we characterize a twisting circle $c$ for which $\lbrace (K_n , r_n) \rbrace$ 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.

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Received 19 August 2015

Accepted 13 April 2017

Published 8 October 2019