On definite strongly quasipositive links and L-space branched covers (1811.08862v2)

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 $\delta_n = \sigma_1 \sigma_2 \ldots \sigma_{n-1}$ is the dual Garside element and $b = \delta_n^k P \in B_n$ is a strongly quasipositive braid whose braid closure $\widehat b$ is definite, then $k \geq 2$ implies that $\widehat b$ 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 $\widehat b$ is an L-space, then $\widehat b$ is one of these links. We show by example that there are strongly quasipositive braids $\delta_n P$ 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.