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

TL;DR

This paper provides a new explicit construction of overtwisted disks in cyclic branched covers of the 3-sphere, demonstrating their existence and location within the contact topology framework.

Contribution

It introduces an explicit method to construct overtwisted disks in cyclic branched covers, offering an alternative proof to a known result in contact topology.

Findings

Explicit construction of overtwisted disks in cyclic branched covers

Demonstration that overtwisted disks lie outside the branch locus

Alternative proof of a key result in contact topology

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.