On the Classification of Legendrian Rational Tangles

TL;DR

This paper investigates how flyping operations on rational tangles can alter their Legendrian isotopy class in contact geometry, revealing distinctions not apparent in topological isotopy, using novel diagramming techniques.

Contribution

It introduces a new diagramming method for rational tangles that helps distinguish Legendrian isotopy classes beyond topological equivalence.

Findings

Flyping can change Legendrian isotopy class without affecting topological class.

New diagramming technique simplifies analysis of Legendrian rational tangles.

Method aids in describing characteristic foliations of compressing discs.

Abstract

We show that under certain conditions the flyping operation on rational tangles, which produces topologically isotopic tangles, may also produce tangles which are not Legendrian isotopic when viewed in the standard contact structure on $\mathbb{R}^3$. This work is motivated by questions posed by Traynor, and incorporates techniques inspired by work of Eliashberg and Fraser. The proofs within employ a new method of diagramming rational tangles which is also used to easily describe the characteristic foliations of compressing discs for their complements.