On definite strongly quasipositive links and L-space branched covers

Abstract

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=σ1σ2…σn−1 is the dual Garside element and b=δnkP∈Bn is a strongly quasipositive braid whose braid closure bˆ is definite, then k≥2 implies that bˆ 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 [7, Theorem 1.1] we deduce that if one of the standard cyclic branched covers of bˆ is an L-space, then bˆ is one of these links. We show by example that there are strongly quasipositive braids δnP 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.