Communications in Analysis and Geometry

Volume 27 (2019)

Number 4

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

Pages: 743 – 790



Kenneth L. Baker (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

Kimihiko Motegi (Department of Mathematics, Nihon University, Tokyo, Japan)


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.

Received 19 August 2015

Accepted 13 April 2017

Published 8 October 2019