Branched covers of quasi-positive links and L-spaces

Let L be an oriented link such that sigma n(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 quasi-positive link other than one with Alexander polynomial a multiple of (t-1)2g(L)+(|L|-1), or L is a quasi-positive 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 quasi-positive knot with monic Alexander polynomial such as an L-space knot, we show that sigma n(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 sigma n(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 quasi-positive links. They also allow us to classify strongly quasi-positive alternating links and 3-strand pretzel links.