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Product structures for Legendrian contact homology

Published online by Cambridge University Press:  02 December 2010

GOKHAN CIVAN ,
PAUL KOPROWSKI ,
JOHN ETNYRE ,
JOSHUA M. SABLOFF  and
ALDEN WALKER
GOKHAN CIVAN
Affiliation:
University of Maryland, College Park, MD 20742, U.S.A.
PAUL KOPROWSKI
Affiliation:
University of Maryland, College Park, MD 20742, U.S.A.
JOHN ETNYRE
Affiliation:
Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. e-mail: [email protected]
JOSHUA M. SABLOFF
Affiliation:
Haverford College, Haverford, PA 19041, U.S.A.
ALDEN WALKER
Affiliation:
California Institute of Technology, Pasadena, CA 91125, U.S.A.
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Abstract

Legendrian contact homology (LCH) is a powerful non-classical invariant of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and non-commutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products – and, more generally, an A∞ structure – on the linearized LCH. We apply the products and A∞ structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 150 , Issue 2 , March 2011 , pp. 291 - 311
Copyright
Copyright © Cambridge Philosophical Society 2010

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References

REFERENCES

[1]Bennequin, D.Entrelacements et equations de Pfaff. Asterisque. 107–108 (1983), 87–161.Google Scholar
[2]Chekanov, Yu.Differential algebra of Legendrian links. Invent. Math. 150 (2002), 441–483.Google Scholar
[3]Chekanov, Yu. and Pushkar, P.Combinatorics of Legendrian links and the Arnol'd 4-conjectures. Russ. Math. Surv. 60 (1) (2005), 95–149.CrossRefGoogle Scholar
[4]Ding, F. and Geiges, H.Legendrian knots and links classified by classical invariants. Commun. Contemp. Math. 9 (2) (2007), 135–162.CrossRefGoogle Scholar
[5]Ekholm, T., Etnyre, J. and Sabloff, J.A duality exact sequence for Legendrian contact homology. Duke Math. J. 150 (1) (2009), 1–75.CrossRefGoogle Scholar
[6]Ekholm, T., Etnyre, J. and Sullivan, M.The contact homology of Legendrian submanifolds in ℝ2n + 1. J. Differential Geom. 71 (2) (2005), 177–305.CrossRefGoogle Scholar
[7]Ekholm, T., Etnyre, J. and Sullivan, M.Non-isotopic Legendrian submanifolds in ℝ2n + 1. J. Differential Geom. 71 (1) (2005), 85–128.Google Scholar
[8]Ekholm, T., Etnyre, J. and Sullivan, M.Orientations in Legendrian contact homology and exact Lagrangian immersions. Internat. J. Math. 16 (5) (2005), 453–532.CrossRefGoogle Scholar
[9]Ekholm, T., Etnyre, J. and Sullivan, M.Legendrian contact homology in P × ℝ. Trans. Amer. Math. Soc. 359 (7) (electronic), (2007), 3301–3335.CrossRefGoogle Scholar
[10]Eliashberg, Ya. Invariants in contact topology. In Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), number Extra Vol. II, pages 327–338 (electronic) (1998).CrossRefGoogle Scholar
[11]Eliashberg, Ya. and Fraser, M. Classification of topologically trivial Legendrian knots. In Geometry, topology and dynamics (Montreal, PQ, 1995), pages 17–51. (Amer. Math. Soc., 1998).Google Scholar
[12]Etnyre, J. Legendrian and transversal knots. In Handbook of Knot Theory, pages 105–185. (Elsevier B. V. 2005).CrossRefGoogle Scholar
[13]Etnyre, J. and Honda, K.Knots and contact geometry I: Torus knots and the figure eight knot. J. Symplectic Geom. 1 (1) (2001), 63–120.Google Scholar
[14]Etnyre, J., Ng, L. and Sabloff, J.Invariants of Legendrian knots and coherent orientations. J. Symplectic Geom. 1 (2) (2002), 321–367.CrossRefGoogle Scholar
[15]Fuchs, D.Chekanov-Eliashberg invariant of Legendrian knots: existence of augmentations. J. Geom. Phys. 47 (1) (2003), 43–65.CrossRefGoogle Scholar
[16]Fuchs, D. and Ishkhanov, T.Invariants of Legendrian knots and decompositions of front diagrams. Mosc. Math. J. 4 (3) (2004), 707–717.Google Scholar
[17]Fuchs, D. and Tabachnikov, S.Invariants of Legendrian and transverse knots in the standard contact space. Topology, 36 (1997), 1025–1053.Google Scholar
[18]Fukaya, K., Oh, Y.-G., Ohta, H. and Ono, K.Lagrangian intersection Floer theory: anomaly and obstruction. Parts I and II. vol. 46. Studies in Advanced Mathematics. (Amer. Math. Soc. 2009).Google Scholar
[19]Kadeišvili, T.On the theory of homology of fiber spaces. Uspekhi Mat. Nauk 35 (3) (1980), 183–188. International Topology Conference (Moscow State University, 1979).Google Scholar
[20]Kajiura, H.Noncommutative homotopy algebras associated with open strings. Rev. Math. Phys. 19 (1) (2007), 1–99.CrossRefGoogle Scholar
[21]Kontsevich, M. and Soibelman, Y. Homological mirror symmetry and torus fibrations. In Symplectic geometry and mirror symmetry (Seoul, 2000), pages 203–263, (World Scientific Publ., 2001).Google Scholar
[22]Lu, D.-M., Palmieri, J. H., Wu, Q.-S. and Zhang, J. J.A-infinity structure on Ext-algebras. J. Pure Appl. Algebra 213 (11) (2009), 2017–2037.CrossRefGoogle Scholar
[23]Markl, M.Transferring A ∞ (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo (2) Suppl. (79) (2006), 139–151.Google Scholar
[24]Melvin, P. and Shrestha, S.The nonuniqueness of Chekanov polynomials of Legendrian knots. Geom. Topol. 9 (2005), 1221–1252.CrossRefGoogle Scholar
[25]Merkulov, S. A.Strong homotopy algebras of a Kähler manifold. Internat. Math. Res. Notices (3) (1999), 153–164.CrossRefGoogle Scholar
[26]Ng, L.Computable Legendrian invariants. Topology 42 (1) (2003), 55–82.CrossRefGoogle Scholar
[27]Ng, L. and Sabloff, J.The correspondence between augmentations and rulings for Legendrian knots. Pacific J. Math. 224 (1) (2006), 141–150.Google Scholar
[28]Ozsváth, P., Szabó, Z. and Thurston, D.Legendrian knots, transverse knots and combinatorial Floer homology. Geom. Topol. 12 (2) (2008), 941–980.CrossRefGoogle Scholar
[29]Rutherford, D. Thurston–Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond. Internat. Math. Res. Notion pages Art. ID 78591, 15pp. (2006).CrossRefGoogle Scholar
[30]Sabloff, J.Augmentations and rulings of Legendrian knots. Internat. Math. Res. Notion (19) (2005), 1157–1180.CrossRefGoogle Scholar
[31]Sabloff, J.Duality for Legendrian contact homology. Geom. Topol. 10 (electronic), (2006), 2351–2381.CrossRefGoogle Scholar
[32]Smirnov, V.Simplicial and operad methods in algebraic topology, volume 198 of Translations of Mathematical Monographs. (Amer. Math. Soc. 2001). Translated from the Russian manuscript by G. L. Rybnikov.CrossRefGoogle Scholar
[33]Stasheff, J.Homotopy associativity of h-spaces. ii. Trans. Amer. Math. Soc. 108 (2) (1963), 293–312.Google Scholar