Abstract: A 3-manifold is called an $L$-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" $3$-manifolds is known. Manifolds obtained as the double-branched cyclic cover of a knot in the $3$-sphere give many examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are $L$-spaces for every index $n$. This is joint work with Ahmad Issa.
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