Transverse knots determined by their cyclic branched covers

TL;DR

This paper explores how cyclic branched covers can distinguish or fail to distinguish transverse knots, providing new examples and conditions for their classification.

Contribution

It constructs new examples of non-isotopic transverse knots with contactomorphic cyclic branched covers and identifies cases where these covers determine the knots.

Findings

Constructed new examples of non-isotopic transverse knots with contactomorphic cyclic branched covers.

Proved that for many transverse knots, their isotopy class is determined by the contactomorphism type of their cyclic branched covers.

Abstract

Harvey-Kawamuro-Plamenevskaya demonstrated the existence of (transversely) non-isotopic transverse knots such that for every $n>1$ their $n$-fold cyclic branched covers are contactomorphic. In this short note, we construct other examples of non-isotopic transverse knots that have contactomorphic cyclic branched covers. Conversely, we prove that the transverse isotopy classes of many transverse knots are actually determined by the contactomorphism type of their cyclic branched covers.