Symmetries of knots, branched cyclic covers, and L-spaces

This dissertation studies the L-space conjecture among manifolds which are branched cyclic covers of links. We present three main results. First, we construct new families of knots all of whose branched cyclic covers are L-spaces. Then, we give an almost complete characterization of which cyclic branched covers of double-twist knots have left-orderable fundamental groups. Finally, we relate the notion of visibility of certain symmetries of an alternating knot to the Heegaard Floer homology of its cyclic branched covers.