Publications mathématiques de l'IHÉS

, Volume 113, Issue 1, pp 97–208

Khovanov homology is an unknot-detector

Article

Abstract

We prove that a knot is the unknot if and only if its reduced Khovanov cohomology has rank 1. The proof has two steps. We show first that there is a spectral sequence beginning with the reduced Khovanov cohomology and abutting to a knot homology defined using singular instantons. We then show that the latter homology is isomorphic to the instanton Floer homology of the sutured knot complement: an invariant that is already known to detect the unknot.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Akbulut, T. Mrowka, and Y. Ruan, Torsion classes and a universal constraint on Donaldson invariants for odd manifolds, Trans. Am. Math. Soc., 347 (1995), 63–76. MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    J. A. Baldwin, On the spectral sequence from Khovanov homology to Heegaard Floer homology, prepint (2008). Google Scholar
  3. 3.
    J. Bloom, A link surgery spectral sequence in monopole Floer homology, prepint (2009). Google Scholar
  4. 4.
    P. J. Braam and S. K. Donaldson, Floer’s work on instanton homology, knots and surgery, in The Floer Memorial Volume, Progr. Math., vol. 133, pp. 195–256, Birkhäuser, Basel, 1995. Google Scholar
  5. 5.
    A. Dold and H. Whitney, Classification of oriented sphere bundles over a 4-complex, Ann. Math. (2), 69 (1959), 667–677. MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. K. Donaldson, The orientation of Yang-Mills moduli spaces and 4-manifold topology, J. Differ. Geom., 26 (1987), 397–428. MathSciNetMATHGoogle Scholar
  7. 7.
    S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, New York, 1990. MATHGoogle Scholar
  8. 8.
    A. Floer, An instanton-invariant for 3-manifolds, Commun. Math. Phys., 118 (1988), 215–240. MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    A. Floer, Instanton homology, surgery, and knots, in Geometry of Low-Dimensional Manifolds, 1, London Math. Soc. Lecture Note Ser., vol. 150, pp. 97–114, Cambridge Univ. Press, Cambridge, 1990. Google Scholar
  10. 10.
    J. E. Grigsby and S. Wehrli, On the colored Jones polynomial, sutured Floer homology, and knot Floer homology, Adv. Math., 223 (2010), 2114–2165. MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    M. Hedden, Khovanov homology of the 2-cable detects the unknot, Math. Res. Lett., 16 (2009), 991–994. MathSciNetMATHGoogle Scholar
  12. 12.
    M. Hedden and L. Watson, Does Khovanov homology detect the unknot? Am. J. Math. (2010). doi:10.1353/ajm.2010.0005. MathSciNetGoogle Scholar
  13. 13.
    A. Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol., 6 (2006), 1429–1457, electronic. MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Juhász, Floer homology and surface decompositions, Geom. Topol., 12 (2008), 299–350. MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    T. Kawasaki, The index of elliptic operators over V-manifolds, Nagoya Math. J., 84 (1981), 135–157. MathSciNetMATHGoogle Scholar
  16. 16.
    M. Khovanov, A categorification of the Jones polynomial, Duke Math. J., 101 (2000), 359–426. MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    P. B. Kronheimer, An obstruction to removing intersection points in immersed surfaces, Topology, 36 (1997), 931–962. MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I, Topology, 32 (1993), 773–826. MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. II, Topology, 34 (1995), 37–97. MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-Manifolds, New Mathematical Monographs, Cambridge University Press, Cambridge, 2007. MATHCrossRefGoogle Scholar
  21. 21.
    P. B. Kronheimer and T. S. Mrowka, Knot homology groups from instantons, preprint (2008). Google Scholar
  22. 22.
    P. B. Kronheimer and T. S. Mrowka, Instanton Floer homology and the Alexander polynomial, Algebr. Geom. Topol., 10 (2010), 1715–1738. MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    P. B. Kronheimer and T. S. Mrowka, Knots, sutures, and excision, J. Differ. Geom., 84 (2010), 301–364. MathSciNetMATHGoogle Scholar
  24. 24.
    P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. Math. (2), 165 (2007), 457–546. MATHCrossRefGoogle Scholar
  25. 25.
    E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math., 197 (2005), 554–586. MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    C. Manolescu, An unoriented skein exact triangle for knot Floer homology, Math. Res. Lett., 14 (2007), 839–852. MathSciNetMATHGoogle Scholar
  27. 27.
    C. Manolescu and P. Ozsváth, On the Khovanov and knot Floer homologies of quasi-alternating links, in Proceedings of Gökova Geometry-Topology Conference 2007 (Gökova Geometry/Topology Conference (GGT), Gökova), pp. 60–81, 2008. Google Scholar
  28. 28.
    P. Ozsváth and Z. Szabó, Holomorphic disks and knot invariants, Adv. Math., 186 (2004), 58–116. MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    P. Ozsváth and Z. Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math., 194 (2005), 1–33. MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    J. Rasmussen, Khovanov homology and the slice genus, Invent. Math. (2010). doi:10.1007/s00222-010-0275-6. MathSciNetGoogle Scholar
  31. 31.
    J. Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University, 2003. Google Scholar
  32. 32.
    J. Rasmussen, Knot polynomials and knot homologies, in Geometry and Topology of Manifolds, Fields Inst. Commun., vol. 47, pp. 261–280, Am. Math. Soc., Providence, 2005. Google Scholar
  33. 33.
    C. H. Taubes, Casson’s invariant and gauge theory, J. Differ. Geom., 31 (1990), 547–599. MathSciNetMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Harvard UniversityCambridgeUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations