Selecta Mathematica

, Volume 18, Issue 2, pp 417–472 | Cite as

Surgery obstructions from Khovanov homology



For a 3-manifold with torus boundary admitting an appropriate involution, we show that Khovanov homology provides obstructions to certain exceptional Dehn fillings. For example, given a strongly invertible knot in S 3, we give obstructions to lens space surgeries, as well as obstructions to surgeries with finite fundamental group. These obstructions are based on homological width in Khovanov homology, and in the case of finite fundamental group depend on a calculation of the homological width for a family of Montesinos links.


Khovanov homology Homological width Twofold branched cover Tangles Dehn surgery Exceptional surgery Finite filling 

Mathematics Subject Classification (2010)

57M12 57M27 57M50 


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  1. 1.
    Bailey J., Rolfsen D.: An unexpected surgery construction of a lens space. Pacific J. Math. 71(2), 295–298 (1977)MathSciNetGoogle Scholar
  2. 2.
    Baker K.L., Grigsby J.E., Hedden M.: Grid diagrams for lens spaces and combinatorial knot Floer homology. Int. Math. Res. Not. IMRN 10(024), 39 (2008)MathSciNetGoogle Scholar
  3. 3.
    Bar-Natan, D., Green, J.: \({{\tt JavaKh}}\). Available at
  4. 4.
    Berge, J.: Some knots with surgeries yielding lens spaces. Unpublished manuscriptGoogle Scholar
  5. 5.
    Bleiler, S.A.: Prime tangles and composite knots. In: Knot theory and manifolds (Vancouver, B.C., 1983), Lecture Notes in Mathematics, vol. 1144, pp. 1–13. Springer, Berlin (1985)Google Scholar
  6. 6.
    Boileau M., Otal J.-P.: Scindements de Heegaard et groupe des homéotopies des petites variétés de Seifert. Invent. Math. 106(1), 85–107 (1991)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Boileau, M., Porti, J.: Geometrization of 3-orbifolds of cyclic type. Astérisque, 272, 208 (2001). Appendix A by Michael Heusener and PortiGoogle Scholar
  8. 8.
    Boyer, S.: Dehn surgery on knots. In: Handbook of geometric topology, pp. 165–218. North-Holland, Amsterdam (2002)Google Scholar
  9. 9.
    Boyer S., Zhang X.: Finite Dehn surgery on knots. J. Am. Math. Soc. 9(4), 1005–1050 (1996)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Culler M., Gordon C.M., Luecke J., Shalen P.B.: Dehn surgery on knots. Ann. Math. 125(2), 237–300 (1987)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Delman C.: Essential laminations and Dehn surgery on 2-bridge knots. Topol. Appl. 63(3), 201–221 (1995)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fintushel R., Stern R.J.: Constructing lens spaces by surgery on knots. Math. Z. 175(1), 33–51 (1980)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Futer D., Ishikawa M., Kabaya Y., Mattman T.W., Shimokawa K.: Finite surgeries on three-tangle pretzel knots. Algebr. Geom. Topol. 9(2), 743–771 (2009)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gordon C.M., Litherland R.A.: On the signature of a link. Invent. Math. 47(1), 53–69 (1978)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Gordon, C.M.: Dehn surgery on knots. In: Proceedings of the International Congress of Mathematicians, vol. I, II (Kyoto, 1990), pp. 631–642, Tokyo, Mathematical Society of Japan (1991).Google Scholar
  16. 16.
    Hedden M.: On Floer homology and the Berge conjecture on knots admitting lens space surgeries. Trans. Am. Math. Soc. 363(2), 949–968 (2011)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hedden M., Watson L.: Does Khovanov homology detect the unknot?. Am. J. Math. 132(5), 1339–1345 (2010)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Heil W.: Elementary surgery on Seifert fiber spaces. Yokohama Math. J. 22, 135–139 (1974)MathSciNetMATHGoogle Scholar
  19. 19.
    Hodgson, C., Rubinstein, J.H.: Involutions and isotopies of lens spaces. In: Knot theory and Manifolds (Vancouver, B.C., 1983), Lecture Notes in Mathematics, vol. 1144, pp. 60–96. Springer, Berlin (1985).Google Scholar
  20. 20.
    Hoste, J., Thistlethwaite, M.: \({{\tt Knotscape}}\). Available at
  21. 21.
    Ichihara K., Jong I.D.: Cyclic and finite surgeries on Montesinos knots. Algebr. Geom. Topol. 9(2), 731–742 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Jones V.F.R.: A polynomial invariant of knots via von Neumann algebras. Bul. Am. Math. Soc. 12, 103–111 (1985)MATHCrossRefGoogle Scholar
  23. 23.
    Khovanov M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Khovanov M.: Patterns in knot cohomology. I. Experiment. Math. 12(3), 365–374 (2003)MathSciNetMATHGoogle Scholar
  25. 25.
    Kirk P.A., Klassen E.P.: Chern-Simons invariants of 3-manifolds and representation spaces of knot groups. Math. Ann. 287(2), 343–367 (1990)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Klassen E.P.: Representations of knot groups in SU(2). Trans. Am. Math. Soc. 326(2), 795–828 (1991)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Lee E.S.: An endomorphism of the Khovanov invariant. Adv. Math. 197(2), 554–586 (2005)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Lickorish W.B.R.: Prime knots and tangles. Trans. Am. Math. Soc. 267(1), 321–332 (1981)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Manolescu, C., Ozsváth, P.: On the Khovanov and knot Floer homologies of quasi-alternating links. In: Proceedings of Gökova geometry-topology conference 2007, pp. 60–81. Gökova Geometry/Topology Conference (GGT), Gökova (2008)Google Scholar
  30. 30.
    Mattman, T.: The Culler-Shaler seminorms of pretzel knots. PhD, McGill University (2000)Google Scholar
  31. 31.
    Montesinos, J.M.: Surgery on links and double branched covers of S 3. In: Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox), pp. 227–259. Annals of Mathematical Studies, No. 84. Princeton University Press, Princeton (1975)Google Scholar
  32. 32.
    Montesinos, J.M.: Revêtements ramifiés de nœds, espaces fibré de Seifert et scindements de Heegaard. Lecture notes, Orsay (1976)Google Scholar
  33. 33.
    Montesinos J.M.: Classical Tessellations and Three-Manifolds. Universitext. Springer, Berlin (1987)CrossRefGoogle Scholar
  34. 34.
    Moser L.: Elementary surgery along a torus knot. Pacific J. Math. 38, 737–745 (1971)MathSciNetMATHGoogle Scholar
  35. 35.
    Osborne R.P.: Knots with Heegaard genus 2 complements are invertible. Proc. Am. Math. Soc. 81(3), 501–502 (1981)MathSciNetMATHGoogle Scholar
  36. 36.
    Ozsváth P., Szabó Z.: Knot Floer homology and the four-ball genus. Geom. Topol. 7, 615–639 (2003) (electronic)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Ozsváth P., Szabó Z.: Holomorphic disks and genus bounds. Geom. Topol. 8, 311–334 (2004) (electronic)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Ozsváth P., Szabó Z.: On knot Floer homology and lens space surgeries. Topology 44(6), 1281–1300 (2005)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Ozsváth P., Szabó Z.: On the Heegaard Floer homology of branched double-covers. Adv. Math. 194(1), 1–33 (2005)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Ozsváth P.S., Szabó Z.: Knot Floer homology and integer surgeries. Algebr. Geom. Topol. 8(1), 101–153 (2008)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Rasmussen J.: Lens space surgeries and a conjecture of Goda and Teragaito. Geom. Topol. 8, 1013–1031 (2004) (electronic)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Rasmussen, J.: Knot polynomials and knot homologies. In: Geometry and topology of manifolds. Fields Inst. Commun., vol. 47, pp. 261–280. American Mathematical Society, Providence (2005)Google Scholar
  43. 43.
    Rasmussen, J.: Lens space surgeries and L-space homology spheres (2007). math.GT/0710.2531Google Scholar
  44. 44.
    Rasmussen J.: Khovanov homology and the slice genus. Invent. Math. 182(2), 419–447 (2010)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Rolfsen D.: Knots and links. Mathematics Lecture Series, No. 7. Publish or Perish Inc., Berkeley (1976)Google Scholar
  46. 46.
    Schubert H.: Knoten mit zwei Brücken. Math. Z. 65, 133–170 (1956)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Scott P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15(5), 401–487 (1983)MATHCrossRefGoogle Scholar
  48. 48.
    Shumakovitch, A.: Torsion of the Khovanov homology (2004). Math.GT/0405474Google Scholar
  49. 49.
    Tanguay, D.: Chirurgie finie et noeuds rationnels. PhD, Université du Québec à Montréal (1996)Google Scholar
  50. 50.
    Thurston, W.P.: The geometry and topology of three-manifolds. Lecture notes (1980)Google Scholar
  51. 51.
    Thurston W.P.: Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Am. Math. Soc. (N.S.) 6(3), 357–381 (1982)MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Waldhausen F.: Über Involutionen der 3-Sphäre. Topology 8, 81–91 (1969)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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