Geometry & Topology

Contact homology and one parameter families of Legendrian knots

Tamas Kalman

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Abstract

We consider S1–families of Legendrian knots in the standard contact R3. We define the monodromy of such a loop, which is an automorphism of the Chekanov–Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S1,3) of Legendrian knots, although it is contractible in the space Emb(S1,3) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid and construct an augmentation for each such link diagram.

Article information

Source
Geom. Topol., Volume 9, Number 4 (2005), 2013-2078.

Dates
Received: 3 October 2004
Revised: 24 July 2005
Accepted: 17 September 2005
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799674

Digital Object Identifier
doi:10.2140/gt.2005.9.2013

Mathematical Reviews number (MathSciNet)
MR2209366

Zentralblatt MATH identifier
1095.53059

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
Legendrian contact homology monodromy Reidemeister moves braid positive knots torus knots

Citation

Kalman, Tamas. Contact homology and one parameter families of Legendrian knots. Geom. Topol. 9 (2005), no. 4, 2013--2078. doi:10.2140/gt.2005.9.2013. https://projecteuclid.org/euclid.gt/1513799674


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References

  • V I Arnol'd, Invariants and perestroikas of fronts on a plane, Trudy Mat. Inst. Steklov. 209 (1995) 14–64
  • C Benham, X-S Lin, D Miller, Subspaces of knot spaces, Proc. Amer. Math. Soc. 129 (2001) 3121–3127
  • D Bennequin, Entrelacements et équations de Pfaff, from: “Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982)”, Astérisque 107, Soc. Math. France, Paris (1983) 87–161
  • F Bourgeois, Contact homology and homotopy groups of the space of contact structures.
  • Y Chekanov, New invariants of Legendrian knots, from: “European Congress of Mathematics, Vol. II (Barcelona, 2000)”, Progr. Math. 202, Birkhäuser, Basel (2001) 525–534
  • Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441–483
  • T Ekholm, J Etnyre, M Sullivan, Legendrian submanifolds in $\R^{2n+1}$ and contact homology.
  • Y Eliashberg, M Fraser, Classification of topologically trivial Legendrian knots, from: “Geometry, topology, and dynamics (Montreal, PQ, 1995)”, CRM Proc. Lecture Notes 15, Amer. Math. Soc., Providence, RI (1998) 17–51
  • Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560–673
  • J B Etnyre, Legendrian and Transversal Knots.
  • J B Etnyre, K Honda, Knots and contact geometry. I. Torus knots and the figure eight knot, J. Symplectic Geom. 1 (2001) 63–120
  • J B Etnyre, K Honda, On connected sums and Legendrian knots, Adv. Math. 179 (2003) 59–74
  • J B Etnyre, K Honda, Cabling and transverse simplicity, e-print.
  • J B Etnyre, L L Ng, J M Sabloff, Invariants of Legendrian knots and coherent orientations, J. Symplectic Geom. 1 (2002) 321–367
  • Y Félix, S Halperin, J-C Thomas, Differential graded algebras in topology, from: “Handbook of algebraic topology”, North-Holland, Amsterdam (1995) 829–865
  • A Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989) 575–611
  • G K Francis, Extensions to the disk of properly nested plane immersions of the circle, Michigan Math. J. 17 (1970) 377–383
  • D Fuchs, Chekanov-Eliashberg invariant of Legendrian knots: existence of augmentations, J. Geom. Phys. 47 (2003) 43–65
  • D Fuchs, T Ishkhanov, Invariants of Legendrian knots and decompositions of front diagrams, Mosc. Math. J. 4 (2004) 707–717, 783
  • A Hatcher, Spaces of knots.
  • M Hutchings, Floer homology of families I \arxivmath.SG/0308115
  • M Hutchings, Floer homology of families II: symplectomorphisms, in preparation
  • L L Ng, Computable Legendrian invariants, Topology 42 (2003) 55–82
  • V Poénaru, Extensions des immersions en codimension $1$ (d'après Samuel Blank), Séminaire Bourbaki, Vol. 10, Exp. No. 342, Soc. Math. France, Paris (1995) 473–505
  • J M Sabloff, Augmentations and rulings of Legendrian knots, Int. Math. Res. Not. (2005) 1157–1180
  • P Seidel, $\pi\sb 1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997) 1046–1095