Inventiones mathematicae

, Volume 211, Issue 3, pp 1149–1200 | Cite as

A complete knot invariant from contact homology

  • Tobias Ekholm
  • Lenhard Ng
  • Vivek Shende


We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.



TE was supported by the Knut and Alice Wallenberg Foundation and the Swedish Research Council. LN was partially supported by NSF Grant DMS-1406371 and a Grant from the Simons Foundation (# 341289 to Lenhard Ng). VS was partially supported by NSF Grant DMS-1406871 and a Sloan Fellowship.


  1. 1.
    Aganagic, M., Ekholm, T., Ng, L., Vafa, C.: Topological strings, D-model, and knot contact homology. Adv. Theor. Math. Phys. 18(4), 827–956 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourgeois, F., Ekholm, T., Eliashberg, Y.: Symplectic homology product via Legendrian surgery. Proc. Natl. Acad. Sci. U.S.A. 108(20), 8114–8121 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourgeois, F., Ekholm, T., Eliashberg, Y.: Effect of Legendrian surgery. Geom. Topol. 16(1), 301–389 (2012) (with an appendix by Sheel Ganatra and Maksim Maydanskiy) Google Scholar
  4. 4.
    Cieliebak, K., Ekholm, T., Latschev, J.: Compactness for holomorphic curves with switching Lagrangian boundary conditions. J. Symplectic Geom. 8(3), 267–298 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cieliebak, K., Ekholm, T., Latschev, J., Ng, L.: Knot contact homology, string topology, and the cord algebra. J. Éc. Polytech. Math. 4, 661–780 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ekholm, T., Etnyre, J., Ng, L., Sullivan, M.: Filtrations on the knot contact homology of transverse knots. Math. Ann. 355(4), 1561–1591 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ekholm, T., Etnyre, J.B., Ng, L., Sullivan, M.G.: Knot contact homology. Geom. Topol. 17(2), 975–1112 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ekholm, T., Etnyre, J., Sullivan, M.: Non-isotopic Legendrian submanifolds in \(\mathbb{R}^{2n+1}\). J. Differ. Geom. 71(1), 85–128 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ekholm, T., Etnyre, J., Sullivan, M.: Orientations in Legendrian contact homology and exact Lagrangian immersions. Int. J. Math. 16(5), 453–532 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ekholm, T., Etnyre, J., Sullivan, M.: Legendrian contact homology in \(P\times \mathbb{R}\). Trans. Am. Math. Soc. 359(7), 3301–3335 (2007) (electronic) Google Scholar
  11. 11.
    Eliashberg, Y., Glvental, A., Hofer, H.: Introduction to symplectic field theory. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds.) Visions in Mathematics. Modern Birkhauser Classics, pp. 560–673. Birkhauser, Basel (2010)Google Scholar
  12. 12.
    Ekholm, T., Honda, K., Kálmán, T.: Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS). arXiv:1212.1519
  13. 13.
    Ekholm, T.: Rational symplectic field theory over \(\mathbb{Z}_2\) for exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 10(3), 641–704 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ekholm, T., Lekili, Y.: Duality between Lagrangian and Legendrian invariants. arXiv:1701.01284
  15. 15.
    Gordon, C., Lidman, T.: Knot contact homology detects cabled, composite, and torus knots. Proc. Am. Math. Soc. (accepted). arXiv:1509.01642
  16. 16.
    Gordon, C.M.A., Luecke, J.: Knots are determined by their complements. J. Am. Math. Soc. 2(2), 371–415 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Higman, G.: The units of group-rings. Proc. Lond. Math. Soc. 2(46), 231–248 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Howie, J., Short, H.: The band-sum problem. J. Lond. Math. Soc. (2) 31(3), 571–576 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1), 37–65 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kontsevich, M.: Symplectic geometry of homological algebra. Available at the author’s webpage (2009)Google Scholar
  21. 21.
    Mishachev, K.: The \(N\)-copy of a topologically trivial Legendrian knot. J. Symplectic Geom. 1(4), 659–682 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nadler, D.: Wrapped microlocal sheaves on pairs of pants. arXiv:1604.00114
  23. 23.
    Nadler, D.: Arboreal singularities. Geom. Topol. 21(2), 1231–1274 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ng, L.: Framed knot contact homology. Duke Math. J. 141(2), 365–406 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ng, L.: Rational symplectic field theory for Legendrian knots. Invent. Math. 182(3), 451–512 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ng, L.: Combinatorial knot contact homology and transverse knots. Adv. Math. 227(6), 2189–2219 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ng, L.: A topological introduction to knot contact homology. In: Contact and Symplectic Topology, volume 26 of Bolyai Soc. Math. Stud., pp. 485–530. János Bolyai Math. Soc., Budapest (2014)Google Scholar
  28. 28.
    Ng, L., Rutherford, D., Shende, V., Sivek, S., Zaslow, E.: Augmentations are sheaves. arXiv:1502.04939
  29. 29.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22(1), 233–286 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shende, V.: The conormal torus is a complete knot invariant. arXiv:1604.03520
  31. 31.
    Sylvan, Z.: On partially wrapped Fukaya categories. arXiv:1604.02540
  32. 32.
    Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math. (2) 87(1), 56–88 (1968)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Institut Mittag-LefflerDjursholmSweden
  3. 3.Department of MathematicsDuke UniversityDurhamUSA
  4. 4.Department of MathematicsUniversity of California, BerkeleyBerkeleyUSA

Personalised recommendations