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Description

We give constraints on when the $n$-fold cyclic branched cover $\Sigma_n(L)$ of a strongly quasipositive link $L$ can be an L-space. In particular we show that if $K$ is a non-trivial L-space knot and $\Sigma_n(K)$ is an L-space then (1) $n \le 5$; (2) if $n$ = 4 or 5 then $K$ is the torus knot $T(2,3)$; (3) if $n$ = 3 then $K$ is either $T(2,3)$ or $T(2,5)$, or $K$ is hyperbolic and has the same Alexander polynomial as $T(2,5)$; (4) if $n$ = 2 then $\Delta_{K}(t)$ is a non-trivial product of cyclotomic polynomials. (This is joint work with Michel Boileau and Steve Boyer.)