Left Orderability of Cyclic Branched Covers of Rational Knots

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August 2023

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A non-trivial group G is left orderable if there is a total ordering < on G such that g < h implies f g < f h for all f, g, h ∈ G. In this dissertation, we study the left orderability of the fundamental groups of cyclic branched covers of the 3-sphere, S3, branched over rational knots. Specifically, the focus is on the three parameter family of rational knots C(2p, 2m, 2n+1) in the Conway notation. This study is motivated by the L-space conjecture of Boyer-Gordon-Watson, which states that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left oderable. A sufficient condition for the fundamental group of the r-th cyclic branched cover of S3 branched over a prime knot to be left orderable was given by Hu in [12]. As an application, Turner determined the left orderability of the fundamental groups of the cyclic branched covers of the rational knots C(2n + 1, 2, 2) for a positive integer n. In Chapters 2 and 3, we generalize Turners results to the rational knots C(2p, 2m, 2n + 1) where p, m, n are integers.

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