The Legendrian knot complement problem

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.

32 pages, 14 figures; V2: Completly rewritten following the referee's advice. The results on non-equivalent Legendrian knots in lens spaces with contactomorphic exteriors contained a gap and are removed

References in corpus (6)

• Transverse Surgery on Knots in Contact 3-Manifolds
• Heegaard Floer homology and knots determined by their complements
• Exceptional Cosmetic surgeries on $S^3$
• Computing the Thurston-Bennequin invariant in open books
• Knot Complement Problem for L-space $\mathbb{Z}H{S}^3$
• Cosmetic contact surgeries along transverse knots and the knot complement problem

Cited by in corpus (6)

• Computing the Thurston-Bennequin invariant in open books
• Non-loose negative torus knots
• Stein traces and characterizing slopes
• Cosmetic contact surgeries along transverse knots and the knot complement problem
• Ribbon disks with the same exterior
• Computing rotation numbers in open books