Invariants of Legendrian Knots and Coherent Orientations

By John B. Etnyre, Lenhard L. Ng, J. Sabloff. 2001

99 citations recorded by the source corpus.

We provide a translation between Chekanov’s combinatorial theory for invariants of Legendrian knots in the standard contact R and a relative version of Eliashberg and Hofer’s contact homology. We use this translation to transport the idea of “coherent orientations” from the contact homology world to Chekanov’s combinatorial setting. As a result, we obtain a lifting of Chekanov’s differential graded algebra invariant to an algebra over Z[t, t] with a full Z grading.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7c82500d2c647e49b2760452f871e650ad0a928a

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Citations in the corpus

Introductory Lectures on Contact Geometry (2001, background): “Shortly after these examples were found an exciting new invariant was discovered [5, 11, 15 ] that allowed one to find many such “non-simple” Legendrian knots in the standard tight contact structure on S3. The situation for transverse knots is not so well understood: it is unknown whether transvers”
Invariants of Legendrian Knots (2003, methodology): “ur homology theory is not oriented. However, the construction described abovecanbe modiﬁed toassociatewith aLegendrianknotLa DGA gradedbyZand having Z[s,s−1] (where deg(s) = m(L)) as a coeﬃcient ring [10]. After reducing the grading to Z/m(L)Z, and applying the homomorphism Z[s,s−1] → Z/2Zsending bo”
Floer homology of families I (2003, background): “sional TQFT and has a number of applications to smooth topology in three and four dimensions, see e.g. [18]. Example 1.4 (contact homology) The relative contacthomologyofChekanov and Eliashberg-Hofer [6, 9] associates to a (generic) Legendrian knot in R3 a 4 chain complex whose chains are generated ”
Morse flow trees and Legendrian contact homology in 1-jet spaces (2005, unknown): “ar X iv :m at h/ 05 09 38 6v 1 [ m at h. SG ] 1 6 Se p 20”
Augmentations and rulings of Legendrian knots (2004, background): “3It is possible to define the algebra over Z[T, T]; see [10].”
Contact homology and one parameter families of Legendrian knots (2004, methodology): “Besides finding new Floer theories, it is also an important and promising direction of research to generalize more and more features of finite-dimensional Morse theory to these new settings.”
The contact homology of Legendrian submanifolds in R2n+1 (2005, background): “In this case one must allow also a “triple point move” see [2, 11].”
Legendrian and Transversal Knots (2003, background): “16 (Etnyre, Ng and Sabloff 2002, [40]).”
Knot and braid invariants from contact homology II (2003, background): “ticular, for Legendrian knots in R3 with the standard contact structure, Chekanov [3] developed a purely combinatorial formulation of relative contact homology; this work was subsequently extended in [9, 18]. Relative contact homology was also studied for knots in circle bundles in [23], and for R2n”
Non-isotopic Legendrian submanifolds in R2n+1 (2005, unknown): “85”
Transverse knots distinguished by knot Floer homology (2007, background): “We note that Legendrian contact homology, which in general is quite good at distinguishing different Legendrian knots, does not distinguish between L1 and −L1; the Z[t] differential graded algebras [15] for L1 and −L1, which in theory could be distinct, are easily shown to be stable tame isomorphic.”
Framed knot contact homology (2004, unknown): “G T ] 2 2 M ay”