Stein traces and characterizing slopes

TL;DR

This paper demonstrates the existence of infinitely many non-isotopic Legendrian knots with equivalent Stein traces in the standard contact 3-sphere, introducing new constructions and exploring the uniqueness of Stein traces.

Contribution

It provides the first example of non-isotopic Legendrian knots sharing the same Stein trace and develops methods for constructing such knots and analyzing their properties.

Findings

Existence of infinite non-isotopic Legendrian knots with equivalent Stein traces

Development of contact annulus twist and Weinstein handlebody equivalences

Results on characterizing slopes and uniqueness of Stein traces

Abstract

We show that there exists an infinite family of pairwise non-isotopic Legendrian knots in the standard contact 3-sphere whose Stein traces are equivalent. This is the first example of such phenomenon. Different constructions are developed in the article, including a contact annulus twist, explicit Weinstein handlebody equivalences, and a discussion on dualizable patterns in the contact setting. These constructions can be used to systematically construct distinct Legendrian knots in the standard contact 3-sphere with contactomorphic (-1)-surgeries and, in many cases, equivalent Stein traces. In addition, we also discuss characterizing slopes and provide results in the opposite direction, i.e. describe cases in which the Stein trace, or the contactomorphism type of an r-surgery, uniquely determines the Legendrian isotopy type.