Skip to main content
SpringerLink
Log in

Knot homology via derived categories of coherent sheaves II, \(\mathfrak{sl}_{m}\) case

  • Published:
Inventiones mathematicae Aims and scope

Abstract

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum \(\mathfrak{sl}_{m}\) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov–Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu’s by homological mirror symmetry.

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

Similar content being viewed by others

References

  1. Bar-Natan, D.: Fast Khovanov homology computations. J. Knot Theory Ramifications 16(3), 243–255 (2007) math.GT/0606318

    Article  MATH  MathSciNet  Google Scholar 

  2. Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties. Preprint (1995) math.AG/9506012 (Max-Planck-Institut für Mathematik, Bonn)

  3. Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14, 535–554 (2001) math.AG/9908027

    Article  MATH  MathSciNet  Google Scholar 

  4. Cautis, S., Kamnitzer, J.: Knot homology via derived categories of coherent sheaves I, sl(2) case. Duke Math. J. 142(3), 511–588 (2008) math.AG/0701194

    Article  MATH  MathSciNet  Google Scholar 

  5. Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997)

    MATH  Google Scholar 

  6. Ginzburg, V.: Perverse sheaves on a loop group and Langlands’ duality. alg-geom/9511007

  7. Haines, T.: Structure constants for Hecke and representation rings. Int. Math. Res. Not. 39, 2103–2119 (2003) math.RT/0304176

    Article  MathSciNet  Google Scholar 

  8. Haines, T.: Equidimensionality of convolution morphisms and applications to saturation problems. Adv. Math. 207(1), 297–327 (2006) math.AG/0501504

    Article  MATH  MathSciNet  Google Scholar 

  9. Horja, R.P.: Derived category automorphisms from mirror symmetry. Duke Math. J. 127, 1–34 (2005) math.AG/0103231

    Article  MATH  MathSciNet  Google Scholar 

  10. Huybrechts, D.: Fourier–Mukai Transforms in Algebraic Geometry. Oxford Math. Monogr. Oxford University Press, Oxford (2006)

  11. Kassel, C.: Quantum Groups. Springer, New York (1995)

    MATH  Google Scholar 

  12. Kawamata, Y.: D-equivalence and K-equivalence. J. Differ. Geom. 61, 147–171 (2002) math.AG/0205287

    MATH  MathSciNet  Google Scholar 

  13. Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (1999) math.QA/9908171

    Article  MathSciNet  Google Scholar 

  14. Khovanov, M.: A functor-valued invariant of tangles. Algebr. Geom. Topol. 2, 665–741 (2002) math.QA/0103190

    Article  MATH  MathSciNet  Google Scholar 

  15. Khovanov, M., Rozansky, L.: Matrix factorizations and link homology. Fundam. Math. 199, 1–91 (2008) math.QA/0401268

    Article  MATH  MathSciNet  Google Scholar 

  16. Khovanov, M., Thomas, R.: Braid cobordisms, triangulated categories, and flag varieties. Homology Homotopy Appl. 9, 19–94 (2007) math.QA/0609335

    MATH  MathSciNet  Google Scholar 

  17. Logvinenko, T.: Derived McKay correspondence via pure-sheaf transforms. Math. Ann. 341(1), 137–168 (2008) math.AG/0606791

    Article  MATH  MathSciNet  Google Scholar 

  18. Lusztig, G.: Singularities, character formulas and a q-analog of weight multiplicities. Astérisque 101–102, 208–229 (1983)

    MathSciNet  Google Scholar 

  19. Manolescu, C.: Link homology theories from symplectic geometry. Adv. Math. 211, 363–416 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. 166(1), 95–143 (2007) math.RT/0401222

    MATH  Google Scholar 

  21. Mirković, I., Vybornov, M.: On quiver varieties and affine Grassmannians of type A. C. R. Math. Acad. Sci. Paris 336(3), 207–212 (2003) math.AG/0206084

    MATH  MathSciNet  Google Scholar 

  22. Murakami, H., Ohtsuki, T., Yamada, S.: HOMFLY polynomial via an invariant of colored plane graphs. Enseign. Math., II. Sér 44(3-4), 325–360 (1998)

    MATH  MathSciNet  Google Scholar 

  23. Namikawa, Y.: Mukai flops and derived categories. J. Reine. Angew. Math. 560, 65–76 (2003) math.AG/0203287

    MATH  MathSciNet  Google Scholar 

  24. Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces. Birkhäuser, Boston Basel Stuttgart (1980)

    MATH  Google Scholar 

  25. Rasmussen, J.: Some differentials on Khovanov–Rozansky homology. math.GT/0607544

  26. Reshetikhin, N., Turaev, V.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127, 1–26 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Rouquier, R.: Categorification of the braid groups. math.RT/0409593

  28. Seidel, P., Smith, I.: A link invariant from the symplectic geometry of nilpotent slices. Duke Math. J. 134(3), 453–514 (2006) math.SG/0405089

    Article  MATH  MathSciNet  Google Scholar 

  29. Sussan, J.: Category O and sl(k) link invariants. math.QA/0701045

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, USA

    Sabin Cautis

  2. Department of Mathematics, University of California, Berkeley, CA, USA

    Joel Kamnitzer

Authors
  1. Sabin Cautis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joel Kamnitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joel Kamnitzer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cautis, S., Kamnitzer, J. Knot homology via derived categories of coherent sheaves II, \(\mathfrak{sl}_{m}\) case. Invent. math. 174, 165–232 (2008). https://doi.org/10.1007/s00222-008-0138-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-008-0138-6

Keywords

Search

Navigation