Augmentations and rulings of Legendrian knots

By J. Sabloff. 2004

92 citations recorded by the source corpus.

A connection between holomorphic and generating family invariants of Legendrian knots is established; namely, that the existence of a ruling (or decomposition) of a Legendrian knot is equivalent to the existence of an augmentation of its contact homology. This result was obtained independently and using different methods by Fuchs and Ishkhanov [12]. Close examination of the proof yields an algorithm for constructing a ruling given an augmentation. Finally, a condition for the existence of an augmentation in terms of the rotation number is obtained.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5eb9b6acce7983a38a725dc06021ade0ad65b52c

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In Geometry and Topology

1. Lenhard L. Ng
2. John B. Etnyre
3. Tam'as K'alm'an
4. T. Ekholm
5. J. Sabloff
6. Roger Casals
7. Dan Rutherford
8. Steven Sivek

In Mathematical Physics

1. John B. Etnyre
2. Lenhard L. Ng
3. T. Ekholm
4. Tam'as K'alm'an
5. J. Sabloff
6. Roger Casals
7. Dan Rutherford
8. Steven Sivek

Citations in the corpus

• Infinitely many Lagrangian fillings (2020, background): “The classification of Legendrian knots and their Lagrangian fillings has been one of the central areas of research in low-dimensional contact topology [17, 18, 36, 47, 49, 50, 51].”
• The Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: The Fuchs conjecture and beyond (2006, background): “In the case of the Kauffman polynomial bound, the equivalent condition was conjectured by Fuchs in [F], and is precisely the existence of an (ungraded) ruling of a front diagram for K. Interestingly enough, this condition is itself known to be equivalent to the existence of an augmentation on the Le”
• Contact homology and one parameter families of Legendrian knots (2004, background): “Theorem 2.12 (Chekanov–Etnyre–Ng–Sabloff) The sum taken when computing ∂(a) for any crossing a is finite, hence ∂ : A → A is well defined.”
• Generating families and Legendrian contact homology in the standard contact space (2008, background): “It is proved in [19] that if R(L) = 0, then there never exists a 2-graded normal ruling.”
• Augmentations are sheaves (2015, background): “ts, with the local minimum placed just to the right of the local maximum appears, in Figure4.4. Here, the two half twists fit together to form what is commonly called a “dip” in the xy-projection, cf. [Sab05]. Associating a Legendrian contact homology DGA to the perturbed m-copy m xz or m f requires ”
• A Duality Exact Sequence for Legendrian Contact Homology (2008, background): “In dimension three, however, the theory of the existence of augmentations has been well-studied (see [13, 14, 21, 24, 25]), while less work has been done in higher dimensions.”
• Legendrian and Transversal Knots (2003, methodology): “In particular we get the first classification of transverse knots in a non-transversely simple knot type. Theorem 5.23 (Etnyre and Honda, [ 35 ]).”