TL;DR
This paper proves that Legendrian knots in the tight 3-sphere are uniquely determined by their exteriors, introduces a new formula for Thurston-Bennequin invariant, and shows this does not extend to Legendrian links.
Contribution
It establishes the uniqueness of Legendrian knots by their exteriors and provides a new computational formula for Thurston-Bennequin invariant.
Findings
Legendrian knots are determined by their exteriors in the tight 3-sphere.
Counterexamples show this is not true for Legendrian links.
A new user-friendly formula for Thurston-Bennequin invariant is introduced.
Abstract
We prove that every Legendrian knot in the tight contact structure of the 3-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3-sphere. On the way a new user-friendly formula for computing the Thurston-Bennequin invariant of a Legendrian knot in a surgery diagram is given.
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