Branched covers of quasi‐positive links and L‐spaces

Let L be a oriented link such that Σₙ(L), the n-fold cyclic cover of S³ branched over L, is an L-space for some n ≥ 2. We show that if either L is a strongly quasipositive link other than one with Alexander polynomial a multiple of (t-1)2g(L) + (|L|-1), or L is a quasipositive link other than one with Alexander polynomial divisible by (t-1)2g₄(L) + (|L|-1), then there is an integer $n(L)$, determined by the Alexander polynomial of L in the first case and the Alexander polynomial of L and the smooth $4$-genus of L, g₄(L), in the second, such that n ≤ n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that Σₙ(K) is not an L-space for n ≥ 6, and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if Σₙ(K) is an L-space for some $n = 2, 3, 4, 5$. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating quasipositive links. They also allow us to classify strongly quasipositive alternating links and $3$-strand pretzel links.