We investigate the problem of characterising the family of strongly
quasipositive links which have definite symmetrised Seifert forms and apply our
results to the problem of determining when such a link can have an L-space
cyclic branched cover. In particular, we show that if δn​=σ1​σ2​…σn−1​ is the dual Garside element and b=δnk​P∈Bn​ is a strongly quasipositive braid whose braid closure b is
definite, then k≥2 implies that b is one of the torus links
T(2,q),T(3,4),T(3,5) or pretzel links P(−2,2,m),P(−2,3,4). Applying
Theorem 1.1 of our previous paper we deduce that if one of the standard cyclic
branched covers of b is an L-space, then b is one of
these links. We show by example that there are strongly quasipositive braids
δn​P whose closures are definite but not one of these torus or pretzel
links. We also determine the family of definite strongly quasipositive
3-braids and show that their closures coincide with the family of strongly
quasipositive 3-braids with an L-space branched cover.Comment: 62 pages, minor revisions, accepted for publication in Adv. Mat