$n$-Fold Cyclic Branched Covers and Overtwisted Contact Structures

Abstract

This article presents an alternate way to prove a result originally proven by Harvey, Kawamuro, and Plamenvskaya in \cite{HaKaPl}. We accomplish this by explicitly constructing an overtwisted disk in the $p$-fold cyclic branched cover of $S^3$ with the standard contact structure branched along a carefully selected transverse knot. Furthermore, by utilizing this construction of the overtwisted disk, we can see that the overtwisted disk is contained in the complement of the branch locus.