Legendrian Non-simple Whitehead Doubles Of The Trefoil

TL;DR

This paper demonstrates that certain Whitehead doubles of the trefoil knot are Legendrian non-simple, expanding previous results on Whitehead doubles of the unknot by employing knot Floer homology and surgery techniques.

Contribution

It extends the classification of Legendrian non-simplicity to Whitehead doubles of the trefoil knot using advanced Floer homology methods.

Findings

Whitehead doubles of the trefoil are Legendrian non-simple

Application of knot Floer homology to contact topology

Use of the distinguished surgery triangle in proofs

Abstract

Ozsv\'ath and Stipsicz showed that some Eliashberg-Chekanov twist knots, which are Whitehead doubles of the unknot, are not Legendrian simple. We extend their result by considering some Whitehead doubles of the trefoil: Using properties of knot Floer homology and the distinguished surgery triangle, we show that this family of knots is Legendrian non-simple in the standard contact 3-sphere.