Publications mathématiques de l'IHÉS

, Volume 113, Issue 1, pp 97–208 | Cite as

Khovanov homology is an unknot-detector

  • P. B. KronheimerEmail author
  • T. S. Mrowka


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.


Modulus Space Spectral Sequence Floer Homology KHOVANOV Homology Hopf Link 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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. MathSciNetzbMATHCrossRefGoogle 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. MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, New York, 1990. zbMATHGoogle Scholar
  8. 8.
    A. Floer, An instanton-invariant for 3-manifolds, Commun. Math. Phys., 118 (1988), 215–240. MathSciNetzbMATHCrossRefGoogle 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. MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    M. Hedden, Khovanov homology of the 2-cable detects the unknot, Math. Res. Lett., 16 (2009), 991–994. MathSciNetzbMATHGoogle 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. MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    T. Kawasaki, The index of elliptic operators over V-manifolds, Nagoya Math. J., 84 (1981), 135–157. MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Khovanov, A categorification of the Jones polynomial, Duke Math. J., 101 (2000), 359–426. MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    P. B. Kronheimer, An obstruction to removing intersection points in immersed surfaces, Topology, 36 (1997), 931–962. MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I, Topology, 32 (1993), 773–826. MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. II, Topology, 34 (1995), 37–97. MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-Manifolds, New Mathematical Monographs, Cambridge University Press, Cambridge, 2007. zbMATHCrossRefGoogle 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. MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    P. B. Kronheimer and T. S. Mrowka, Knots, sutures, and excision, J. Differ. Geom., 84 (2010), 301–364. MathSciNetzbMATHGoogle Scholar
  24. 24.
    P. Kronheimer, T. Mrowka, P. Ozsváth, and Z. Szabó, Monopoles and lens space surgeries, Ann. Math. (2), 165 (2007), 457–546. zbMATHCrossRefGoogle Scholar
  25. 25.
    E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math., 197 (2005), 554–586. MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    C. Manolescu, An unoriented skein exact triangle for knot Floer homology, Math. Res. Lett., 14 (2007), 839–852. MathSciNetzbMATHGoogle 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. MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    P. Ozsváth and Z. Szabó, On the Heegaard Floer homology of branched double-covers, Adv. Math., 194 (2005), 1–33. MathSciNetzbMATHCrossRefGoogle 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. MathSciNetzbMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag 2011

Authors and Affiliations

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

Personalised recommendations