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Branched covers of quasipositive links and L-spaces

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HTLW02 - 3-manifold workshop

Co-authors: Michel Boileau (Université Aix-Marseille), Cameron McA. Gordon (University of Texas at Austin)
 
We show that if L is an oriented non-trivial strongly quasipositive link or an oriented quasipositive link which does not bound a smooth planar surface in the 4-ball, then the Alexander polynomial and signature function of L determine an integer n(L) such that \Sigma_n(L), the n-fold cyclic cover of S^3 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 \Sigma_n(K) is not an L-space for n \geq 6 and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if \Sigma_n(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 oriented quasipositive links. They also allow us to classify strongly quasipositive 3-strand pretzel knots. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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