On Transverse Knots and Branched Covers

TL;DR

This paper investigates contact manifolds formed as cyclic branched covers of transverse knots in the standard contact 3-sphere, analyzing their properties and equivalences through open books and contact surgeries.

Contribution

It demonstrates that many branched covers are contactomorphic for isotopic transverse knots with identical self-linking numbers, covering a broad class of non-transversely simple knots.

Findings

Branched covers often contactomorphic for isotopic knots with same self-linking.

Describes contact manifolds via open books and surgeries.

Includes most non-transversely simple knots of certain classes.

Abstract

We study contact manifolds that arise as cyclic branched covers of transverse knots in the standard contact 3-sphere. We discuss properties of these contact manifolds and describe them in terms of open books and contact surgeries. In many cases we show that such branched covers are contactomorphic for smoothly isotopic transverse knots with the same self-linking number. These pairs of knots include most of the non-transversely simple knots of Birman-Menasco and Ng-Ozsvath-Thurston.