Skip to main content
SpringerLink
Log in

Khovanov–Seidel quiver algebras and bordered Floer homology

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard–Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Notes

  1. This extra grading has a natural interpretation on the Khovanov side in terms of \(U_q(\mathfrak sl _2)\) weight space decompositions and on the Heegaard–Floer side in terms of relative Spin\(^c\) structures. See [15] for more details.

  2. The difference in shift conventions for homological versus internal gradings is unfortunate, but standard in the literature. In particular, they coincide with those in [23]. The reader should be warned, however, that [31] uses a different convention, since their differential maps decrease rather than increase homological grading.

  3. This internal grading corresponds to the second of the two gradings discussed in Notation 3.2. Note that this grading is not the grading by path length which appears in [11, 50] and corresponds to the \(j\) (quantum) grading of [21]. See Remark 3.21.

  4. We expect that results similar to those described in Theorems 5.1 and 6.1 hold for other choices of basis, but we do not address that here.

  5. In the language of [10], \(Q_j\) is a projective resolution of the standard module associated to the length \(m+1\) weight \(\lambda = (\vee \cdots \vee \wedge \vee \cdots \vee )\), where the lone \(\wedge \) is in the \((j \in \{0,\ldots , m\})\)th position.

  6. Note that although [32, Prop. 2.4.10] is formulated for categories of \(A_{\infty }\) right modules, similar statements also hold for categories of \(A_{\infty }\) left modules and \(A_{\infty }\) bimodules; see [32] for details.

  7. Here we use the notation convention from [54], which differs by a shift from the one in [31]. See the note in [54, Sec. 2.2].

  8. The only non-trivial check that must be performed is that \(\mathcal F _0 \cdot \mathcal F _0 \subseteq \mathcal F _0\), but this follows from the fact that \(\partial _\tau \) satisfies the Leibniz rule.

References

  1. Asaeda, M.M., Przytycki, J.H., Sikora, A.S.: Categorification of the Kauffman bracket skein module of \(I\)-bundles over surfaces. Algebr. Geom. Topol. 4,1177–1210 (2004) [electronic]

    Google Scholar 

  2. Auroux, D.: Fukaya categories and bordered Heegaard–Floer homology. In: Proceedings of International Congress of Mathematicians, vol. II (Hyderabad, 2010), pp. 917–941. Hindustan Book Agency, New Delhi, (2010). math.GT/1003.2962

  3. Auroux, D.: Fukaya categories of symmetric products and bordered Heegaard–Floer homology. J. Gökova Geom. Topol. GGT 4, 1–54, (2010). math.GT/1001.4323

  4. Auroux, D., Grigsby, J.E., Wehrli, S.M.: Sutured Khovanov homology, Hochschild homology, and the Ozsváth-Szabó spectral sequence (in preparation)

  5. Baldwin, J.A.: Heegaard Floer homology and genus one, one boundary component open books. Topology 1(4), 963–992 (2008)

    Google Scholar 

  6. Baldwin, J.A.: On the spectral sequence from Khovanov homology to Heegaard Floer homology. Int. Math. Res. Not. IMRN 15, 3426–3470 (2011)

    Google Scholar 

  7. Baldwin, J.A., Plamenevskaya, O.: Khovanov homology, open books, and tight contact structures. Adv. Math. 224(6), 2544–2582 (2010)

    Google Scholar 

  8. Berglund, A.: A-infinity algebras and homological perturbation theory (2010)

  9. Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors. Lecture Notes in Mathematics, vol. 1578 Springer, Berlin (1994)

  10. Brundan, J., Stroppel, C.: Highest weight categories arising from Khovanov’s diagram algebra I: cellularity. Mosc. Math. J. 11(4), 685–722 (2011)

    Google Scholar 

  11. Chen, Y., Khovanov, M.: An invariant of tangle cobordisms via subquotients of arc rings (2006). math.QA/0610054

  12. Grigsby, J.E., Wehrli, S.M.: Khovanov homology, sutured Floer homology and annular links. Algebr. Geom. Topol 10(4), 2009–2039 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grigsby, J.E., Wehrli, S.M.: On the colored Jones polynomial, sutured Floer homology, and knot Floer homology. Adv. Math. 223(6), 2114–2165 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grigsby, J.E., Wehrli, S.M.: On the naturality of the spectral sequence from Khovanov homology to Heegaard Floer homology. Int. Math. Res. Not. IMRN (21), 4159–4210 (2010)

  15. Grigsby, J.E., Wehrli, S.M.: On gradings in Khovanov homology and sutured Floer homology. In: Topology and Geometry in Dimension Three. Contemporary Mathematics, vol. 560, pp. 111–128. American Mathematical Society, Providence (2011)

  16. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  17. Juhász, A.: Holomorphic discs and sutured manifolds. Algebr. Geom. Topol. 6, 1429–1457 (2006) [electronic]

    Google Scholar 

  18. Kadeishvili, T.V.: The algebraic structure in the homology of an \(A(\infty )\)-algebra. Soobshch. Akad. Nauk Gruzin. SSR 108(2), 249–252 (1983). 1982

    Google Scholar 

  19. Keller, B.: Introduction to \(A\)-infinity algebras and modules. Homol. Homot. Appl. 3(1), 1–35 (2001)

  20. Keller, B.: \(A\)-infinity algebras, modules and functor categories. In: Trends in Representation Theory of Algebras and Related Topics. Contemporary Mathematics, vol. 406, pp. 67–93. American Mathematical Society, Providence (2006)

  21. Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Khovanov, M.: A functor-valued invariant of tangles. Alg. Geom. Topol. 2, 665–741 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203–271 (2002) [electronic]

    Google Scholar 

  24. Klamt, A.: \(A_\infty \) Structures on the Algebra of Extensions of Verma Modules in the Parabolic Category O. Diploma Thesis, Rheinischen Friedrich–Wilhelms-Universität Bonn (2010). math.RT/1104.0102

  25. Klamt, A., Stroppel, C.: On the Ext algebras of parabolic Verma modules and \(A_\infty \)-structures. J. Pure Appl. Algebra 216(2), 323–336 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kontsevich, M.: Homological Algebra of Mirror Symmetry. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Zürich, 1994), pp. 120–139, Birkhäuser, Basel (1995)

  27. Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. In: Symplectic Geometry and Mirror Symmetry (Seoul, 2000), pp. 203–263. World Scientific Publishing, River Edge (2001)

  28. Kronheimer, P.B., Mrowka, T.S.: Khovanov homology is an unknot-detector. Publ. Math. Inst. Hautes Études Sci. 113, 97–208 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Lefèvre-Hasegawa, K.: Sur les \(A_{\infty }\)-catégories. PhD thesis, Université Paris 7, Paris (2003). math.CT/0310337

  30. Lekili, Y., Perutz, T.: Fukaya categories of the torus and dehn surgery. PNAS 108(2), 8106–8113 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  31. Lipshitz, R., Ozsváth, P., Thurston, D: Bordered Heegaard Floer homology: invariance and pairing (2008). math.GT/0810.0687

  32. Lipshitz, R., Ozsváth, P., Thurston, D.: Bimodules in bordered Heegaard Floer homology (2010). math.GT/1003.0598

  33. Lipshitz, R., Ozsváth, P., Thurston, D.: Heegaard Floer homology as morphism spaces (2010). math.GT/1005.1248

  34. Lipshitz, R., Ozsváth, P.S., Thurston, D.P.: Bordered Floer homology and the spectral sequence of a branched double cover I (2010). math.GT/1011.0499

  35. Lipshitz, R., Ozsváth, P.S., Thurston, D.P.: A faithful linear-categorical action of the mapping class group of a surface with boundary (2010). math.GT/1012.1032

  36. Loi, A., Piergallini, R.: Compact Stein surfaces with boundary as branched covers of B\(^4\). Invent. Math. 143, 325–248 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  37. Manolescu, C., Ozsváth, P., Sarkar, S.: A combinatorial description of knot Floer homology. Ann. Math. 169(2), 633–660 (2009)

    Article  MATH  Google Scholar 

  38. Markl, M.: Transferring \(A_\infty \) (strongly homotopy associative) structures. Rend. Circ. Mat. Palermo (2) Suppl. (79), 139–151 (2006)

  39. Merkulov, S.A.: Strong homotopy algebras of a Kähler manifold. Int. Math. Res. Notices (3), 153–164 (1999)

  40. Ozsváth, P., Szabó, Z.: Heegaard Floer homology and contact structures. Duke Math. J. 129(1), 39–61 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ozsváth, P., Szabó, Z.: On the Heegaard Floer homology of branched double-covers. Adv. Math. 194(1), 1–33 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  42. Plamenevskaya, O.: Transverse knots and Khovanov homology. Math. Res. Lett. 13(4), 571–586 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  43. Roberts, L.P.: On knot Floer homology in double branched covers (2007). math.GT/0706.0741

  44. Roberts, L.P.: Notes on the Heegaard-Floer link surgery spectral sequence (2008). math.GT/0808.2817

  45. Seidel, P.: Vanishing cycles and mutation. In: European Congress of Mathematics, Vol. II (Barcelona, 2000), progress in Mathematics, vol. 202, pp. 65–85. Birkhäuser, Basel (2001)

  46. Seidel, P.: A long exact sequence for symplectic Floer cohomology. Topology 42(5), 1003–1063 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  47. Seidel, P.: Fukaya categories and Picard-Lefschetz theory. In: Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008)

  48. Seidel, P.: Fukaya \(A_\infty \) structures associated to Lefschetz fibrations I (2009). math.SG/0912.3932

  49. Seidel, P., Smith, I.: Localization for involutions in Floer cohomology. Geom. Funct. Anal. 20(6), 1464–1501 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  50. Stroppel, C.: Parabolic category \(\cal {O}\), perverse sheaves on Grassmanians, Springer fibers and Khovanov homology. Compos. Math., pp. 954–992 (2009)

  51. Szabó, Z.: A geometric spectral sequence in Khovanov homology (2010). math.GT/1010.4252

  52. The Knot Atlas. http://katlas.org/wiki/10_124

  53. Watson, L.: Surgery obstructions from Khovanov homology (2008). math.GT/0807.1341

  54. Zarev R.: Bordered Floer homology for sutured manifolds (2009). math.GT/0908.1106

Download references

Acknowledgments

We are grateful to Tony Licata, Robert Lipshitz, Peter Ozsváth, Catharina Stroppel, and Dylan Thurston for a great number of interesting conversations, and to the MSRI semester-long program on Homology Theories of Knots and Links for making these conversations possible. We would also like to thank Joshua Sussan for bringing to our attention that some of the algebraic results of Sect. 3 (in particular, Lemma 3.12) were independently obtained by Angela Klamt and Catharina Stroppel in [24] and [25]. Many thanks are also due to the excellent referee and editor, whose insightful suggestions greatly improved the manuscript. Finally, we are indebted to John Baldwin, who helped us find the example described in Sect. 7.

Author information

Authors and Affiliations

  1. Department of Mathematics, UC Berkeley, 970 Evans Hall # 3840, Berkeley, CA, 94720, USA

    Denis Auroux

  2. Mathematics Department, Boston College, 301 Carney Hall, Chestnut Hill, MA, 02467, USA

    J. Elisenda Grigsby

  3. Mathematics Department, Syracuse University, 215 Carnegie, Syracuse, NY, 13244, USA

    Stephan M. Wehrli

Authors
  1. Denis Auroux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Elisenda Grigsby
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephan M. Wehrli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Elisenda Grigsby.

Additional information

DA was partially supported by NSF grant DMS-1007177. JEG was partially supported by a Viterbi-endowed MSRI postdoctoral fellowship and NSF grant number DMS-0905848. SMW was partially supported by an MSRI postdoctoral fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Auroux, D., Grigsby, J.E. & Wehrli, S.M. Khovanov–Seidel quiver algebras and bordered Floer homology. Sel. Math. New Ser. 20, 1–55 (2014). https://doi.org/10.1007/s00029-012-0106-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-012-0106-2

Keywords

Mathematics Subject Classification (1991)

Search

Navigation