2018/11/21 by Steven Boyer, Boileau, Michel, Boyer, Steven +1 · 1 citation
Mathematics · #57M25 #57M50 #57M99 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · doi:10.48550/arxiv.1811.08862
openalex publication_date 2018/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \widehat b is definite, then k ≥ 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 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.