Geometry and Topology Seminar, "Branched Covers of Quasipositive Links and L-Spaces", Steven Boyer
Département de mathématiques
Université du Québec à Montréal
Title: Branched Covers of Quasipositive Links and L-Spaces.
Abstract: We show that if L is an oriented strongly quasipositive link other than the trivial knot or a link whose Alexander polynomial is a positive power of (t-1), or a quasipositive link with non-zero smooth 4-ball genus, then the Alexander polynomial and signature function of L determine an integer n(L) ≥ 1 such that Σn(L), the n-fold cyclic cover of S3 branched over L, is not an L-space for n > n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that Σn(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 Σn(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 quasipositive links. They also allow us to classify strongly quasipositive alternating links and strongly quasipositive alternating 3-strand pretzel links. This is joint work with Michel Boileau and Cameron Gordon