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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 termsof the linking number $\omega$ between $K$ and $c$. We establish the conjecturein case of $|\omega| \neq 1$, prove that $\{K_n\}$ contains at most threeL-space knots if $\omega = 0$, and address the case where $|\omega| = 1$ underan additional hypothesis about Seifert surgeries. To that end, we characterizea twisting circle $c$ for which $\{ (K_n, r_n) \}$ contains at least tenSeifert surgeries. We also pose a few questions about the nature of twistfamilies of L-space knots, their expressions as closures of positive (ornegative) braids, and their wrapping about the twisting circle.

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