Advertisement

Selecta Mathematica

, Volume 24, Issue 2, pp 1315–1390 | Cite as

Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

  • András Juhász
  • Marco Marengon
Article

Abstract

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a \((1+1)\)-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.

Keywords

Knot cobordism Heegaard Floer homology Khovanov homology Spectral sequence TQFT 

Mathematics Subject Classification

57M27 57R58 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank John Baldwin, Matt Hedden, Tom Hockenhull, Joan Licata, Andrew Lobb, Ciprian Manolescu, Tom Mrowka, Jacob Rasmussen, Ian Zemke, and the anonymous referee for their comments and suggestions. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 674978). The first author was supported by a Royal Society Research Fellowship. The second author was supported by an EPSRC Doctoral Training Award and LMS Grant PMG 16-17 07. The first author would also like to thank the Isaac Newton Institute for its hospitality.

References

  1. 1.
    Arone, G., Kankaanrinta, M.: On the functoriality of the blow-up construction. Bull. Belg. Math. Soc. Simon Stevin 17(5), 821–832 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Atiyah, M.: Topological quantum field theories. Publ. Math. IHÉS 68(1), 175–186 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baldwin, J., Hedden, M., Lobb, A.: On the functoriality of Khovanov–Floer theories. arXiv:1509.04691 (2015)
  4. 4.
    Bar-Natan, D.: On Khovanov’s categorification of the Jones polynomial. Algebr. Geom. Topol. 2(1), 337–370 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bar-Natan, D.: Khovanov’s homology for tangles and cobordisms. Geom. Topol. 9(3), 1443–1499 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Blanchet, C., Turaev, V.: Axiomatic approach to topological quantum field theory. Encycl. Math. Phys. 1, 232–234 (2006)Google Scholar
  7. 7.
    Gabai, D.: Foliations and the topology of 3-manifolds. J. Differ. Geom. 18(3), 445–503 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gabai, D.: Detecting fibred links in \(S^3\). Comment. Math. Helv. 61, 519–555 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Honda, K., Kazez, W., Matić, G.: Contact structures, sutured Floer homology and TQFT. arXiv:0807.2431 (2008)
  10. 10.
    Honda, K., Kazez, W., Matić, G.: The contact invariant in sutured Floer homology. Invent. Math. 176(3), 637–676 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hedden, M., Ni, Y.: Khovanov module and the detection of unlinks. Geom. Topol. 17(5), 3027–3076 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Honda, K.: On the classification of tight contact structures. II. J. Differ. Geom 55(1), 83–143 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jacobsson, M.: An invariant of link cobordisms from Khovanov homology. Algebr. Geom. Topol. 4, 1211–1251 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Juhász, A., Marengon, M.: Concordance maps in knot Floer homology. Geom. Topol. 20(6), 3623–3673 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Juhász, A., Thurston, D.P.: Naturality and mapping class groups in Heegaard Floer homology. arXiv:1210.4996 (2012)
  16. 16.
    Juhász, A.: Holomorphic discs and sutured manifolds. Algebr. Geom. Topol. 6, 1429–1457 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Juhász, A.: Floer homology and surface decompositions. Geom. Topol. 12(1), 299–350 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Juhász, A.: The sutured Floer homology polytope. Geom. Topol. 14, 1303–1354 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Juhász, A.: Defining and classifying TQFTs via surgery. arxiv:1408.0668 (2014)
  20. 20.
    Juhász, A.: Cobordisms of sutured manifolds and the functoriality of link Floer homology. Adv. Math. 299, 940–1038 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Khovanov, M.: Patterns in knot cohomology, I. Exper. Math. 12(3), 365–374 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kronheimer, P., Mrowka, T.: Khovanov homology is an unknot-detector. Publ. Math. IHÉS 113(1), 97–208 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Krcatovich, D.: The reduced knot Floer complex. Topol. Appl. 194, 171–201 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lipshitz, R.: A cylindrical reformulation of Heegaard Floer homology. Geom. Topol. 10(2), 955–1096 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lipshitz, R.: Heegaard Floer Homologies: lecture notes, Lectures on quantum topology in dimension three, Panoramas et synthèses, vol. 48, Société Mathématique de France, pp. 131–174 (2016)Google Scholar
  27. 27.
    Lutz, R.: Structures de contact sur les fibrés principaux en cercles de dimension trois. Ann. Inst. Fourier 27(3), 1–15 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Manolescu, C.: An unoriented skein exact triangle for knot Floer homology. Math. Res. Lett. 14(5), 829–852 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    McCleary, J.: A User’s Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  30. 30.
    Manolescu, C., Ozsváth, P.: On the Khovanov and knot Floer homologies of quasi-alternating links. Proc. Gökova Geometry-Topol. Conf. 2008, 60–81 (2007)zbMATHGoogle Scholar
  31. 31.
    Ozsváth, P., Szabó, Z.: Holomorphic disks and knot invariants. Adv. Math. 186(1), 58–116 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ozsváth, P., Szabó, Z.: On the Heegaard Floer homology of branched double-covers. Adv. Math. 194(1), 1–33 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ozsváth, P., Szabó, Z.: Holomorphic triangles and invariants for smooth four-manifolds. Adv. Math. 202(2), 326–400 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Ozsváth, P., Szabó, Z.: Holomorphic disks, link invariants and the multi-variable Alexander polynomial. Algebr. Geom. Topol. 8(2), 615–692 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rasmussen, J.: Floer homology and knot complements. Ph.D. Thesis, Harvard University (2003)Google Scholar
  36. 36.
    Rasmussen, J.: Knot polynomials and knot homologies. Geom. Topol. Manifolds 47, 261–280 (2005)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Sarkar, S.: Moving basepoints and the induced automorphisms of link Floer homology. Algebr. Geom. Topol. 15, 2479–2515 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wong, C.-M.M.: Grid diagrams and Manolescu’s unoriented skein exact triangle for knot Floer homology. arXiv:1305.2562 (2013)
  39. 39.
    Zemke, I.: Link cobordisms and functoriality in link Floer homology. arXiv:1610.05207 (2016)
  40. 40.
    Zemke, I.: Quasi-stabilization and basepoint moving maps in link Floer homology. arXiv:1604.04316 (2016)
  41. 41.
    Zemke, I.: Link cobordisms and absolute gradings on link Floer homology. arXiv:1701.03454 (2017)

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK
  2. 2.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA

Personalised recommendations