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

摘要

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 the case of vertical bar omega vertical bar not equal 1, prove that {K-n} contains at most three L-space knots if omega = 0, and address the case where vertical bar omega vertical bar = 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.