Generating families and Legendrian contact homology in the standard contact space
By Dmitry Fuchs, Dan Rutherford. 2008
51 citations recorded by the source corpus.
We show that if a Legendrian knot in a standard contact space ℝ3 possesses a generating family, then there exists an augmentation of the Chekanov–Eliashberg differential graded algebra so that the associated linearized contact homology (LCH) is isomorphic to singular homology groups arising from the generating family. In this setting, we show that Sabloff's duality result for LCH may be viewed as Alexander duality. In addition, we provide an explicit construction of a generating family for a front diagram with graded normal ruling and give a new approach to augmentation implies normal ruling.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/809030b9c82284800726c717307939f739970341
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In Computational Theory and Mathematics
- Steven Sivek
- Baptiste Chantraine
- Sheila Sandon
- Lenhard L. Ng
- Dan Rutherford
- Georgios Dimitroglou Rizell
- Eric Zaslow
- Vivek V. Shende
In Geometry and Topology
- Baptiste Chantraine
- Steven Sivek
- Lenhard L. Ng
- Georgios Dimitroglou Rizell
- Vivek V. Shende
- Dan Rutherford
- Frederick J. Bourgeois III
- Sheila Sandon
In Mathematical Physics
- Baptiste Chantraine
- Steven Sivek
- Georgios Dimitroglou Rizell
- Lenhard L. Ng
- Frederick J. Bourgeois III
- Sheila Sandon
- Vivek V. Shende
- Dan Rutherford
Citations in the corpus
- Bilinearised Legendrian contact homology and the augmentation category (2012, background): “For instance, when M = R, it has been shown in [15] that LCHε(Λ) ≃ GF (Λ, F ε) where F ε is an equivalence class of generating family associated to ε via graded ruling.”
- Legendrian ambient surgery and Legendrian contact homology (2012, background): “the case when there exists a generating family for L⊂ J1(M), it is expected that there is an augmentation which induces a complex equivalent to that induced by the generating family. This is shown in [FR] when the contact manifold is the standard contact three-space J1(R). Forthcoming work by Bourge”
- Contact Homology, Capacity and Non-Squeezing in R^2n x S^1 via Generating Functions (2009, methodology): “6], [Sik87], [Giv90], [Vit92], [Tr94], [Giv95], [Chap95],[Th95], [Vit96], [Chek96], [EG98], [Bh98], [Th98], [Mil99], [Th99], [Bh01], [Tr01], [CP05], [FP06], [JT06], [CFP07], [CN08], [CN09], [ELST08], [FR08]). In particular, Viterbo [Vit92] applied Morse-theoretical methods to the generating function”
- Lagrangian cobordisms via generating families: Construction and geography (2014, unknown): “The set of all generating family cohomology groups taken over all possible generating families for ƒ forms an invariant of ƒ up to Legendrian isotopy (see [23; 40]).”
- Augmentations are sheaves (2015, result): “ Lagrangians and augmentations. These are summarized in Section8. We remark that this theorem has already been applied by the third author [She15] to prove a conjecture of Fuchs and the second author [FR11], which asserts that the generating family homologies of a Legendrian knot [Tra01,FR11] coinci”