Branched covers of quasipositive links and L-spaces

Abstract

Let $L$ be a oriented link such that ${\Sigma }_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n\ge 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander polynomial a multiple of $(t-1{)}^{2g(L)+(|L|-1)}$, or $L$ is a quasipositive link other than one with Alexander polynomial divisible by $(t-1{)}^{2g_4(L)+(|L|-1)}$, then there is an integer $n(L)$, determined by the Alexander polynomial of $L$ in the first case and the Alexander polynomial of $L$ and the smooth $4$-genus of $L$, $g_4(L)$, in the second, such that $n\le n(L)$. 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 $n\ge 6$, 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 quasipositive links. They also allow us to classify strongly quasipositive alternating links and $3$-strand pretzel links.

49 pages, 7 figures, minor corrections and improved exposition, accepted for publication by the Journal of Topology

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• Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil
• Stability and triviality of the transverse invariant from Khovanov homology
• A unified Casson-Lin invariant for the real forms of SL(2)
• Tight fibred knots without L-space surgeries
• Taut foliations, braid positivity, and unknot detection
• Circular orderability of 3-manifold groups
• An instanton take on some knot detection results
• Small Heegaard genus and SU(2)