Algebraic & Geometric Topology

A functor-valued invariant of tangles

Mikhail Khovanov

Full-text: Open access


We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle. When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.

Article information

Algebr. Geom. Topol., Volume 2, Number 2 (2002), 665-741.

Received: 21 February 2002
Accepted: 25 April 2002
First available in Project Euclid: 21 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 16D20: Bimodules 18G60: Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]

tangles Jones polynomial Kauffman bracket TQFT complexes bimodules


Khovanov, Mikhail. A functor-valued invariant of tangles. Algebr. Geom. Topol. 2 (2002), no. 2, 665--741. doi:10.2140/agt.2002.2.665.

Export citation


  • L. Abrams. Two-dimensional topological quantum field theories and Frobenius algebras. J. Knot Theory and its Ramifications, 5:569–589, 1996.
  • S. Ariki. On the decomposition numbers of the Hecke algebra of ${G}(m,1,n)$. J. Math. Kyoto Univ., 36(4):789–808, 1996.
  • V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko. Singularities of differentiable maps, vol.I. Monographs in Mathematics, 82. Burkhäuser, Boston, 1985.
  • M. F. Atiyah. Topological quantum field theories. I.H.E.S. Publ. Math., 68:175–186, 1988.
  • J. C. Baez and J. Dolan. Higher-dimensional algebra and topological quantum field theory. J. Math. Phys., 36(11):6073–6105, 1995.
  • B. Bakalov and A. A. Kirillov, Jr. Lectures on tensor categories and modular functors. University Lecture Series 21. AMS, Providence, RI, 2001. A preliminary version is available at$\tilde{\hspace{0.05in}}$kirillov.
  • A. J. Berkson. The u-algebra of a restricted Lie algebra is Frobenius. Proc. Amer. Math. Soc., 15:14–15, 1964.
  • J. Bernstein, I. B. Frenkel, and M. Khovanov. A categorification of the Temperley-Lieb algebra and Schur quotients of ${U}(\mf{sl}_2)$ via projective and Zuckerman functors. Selecta Math., New Ser., 5:199–241, 1999.\nl arXiv:math.QA/0002087
  • J. Bernstein and S. Gelfand. Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras. Compositio Math., 41(2):245–285, 1980.
  • A. Bondal and D. Orlov. Semiorthogonal decompositions for algebraic varieties. arXiv:alg-geom/9506012
  • P. J. Braam and S. K. Donaldson. Floer's work on instanton homology, knots and surgery. In The Floer memorial volume, Progr. Math., 133, pages 195–256. Birkhäuser, Basel, 1995.
  • T. Braden. Perverse sheaves on Grassmannians. arXiv:math.AG/9907152.
  • T. Braden and M. Khovanov. In preparation.
  • T. Bridgeland. Equivalences of triangulated categories and Fourier-Mukai transforms. Bull. London Math. Soc., 31:25–34, 1999. arXiv:math.AG/9809114
  • J. S. Carter, D. E. Flath, and M. Saito. The classical and quantum 6$j$-symbols. Mathematical Notes, 43. Princeton University Press, Princeton, NJ, 1995.
  • J. S. Carter, L. H. Kauffman, and M. Saito. Diagrammatics, singularities and their algebraic interpretations. In 10th Brazilian Topology Meeting (São Carlos, 1996), Mat. Contemp. 13, pages 21–115, 1997.
  • R. W. Carter. Representation theory of the 0-Hecke algebra. Journal of algebra, 104:89–103, 1986.
  • L. Crane and I. B. Frenkel. Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. Journal of Mathematical Physics, 35(10):5136–5154, 1994.
  • R. Dijkgraaf and E. Witten. Topological gauge theories and group cohomology. Comm. Math. Phys., 129(2):393–429, 1990.
  • R. Farnsteiner and H. Strade. Modular Lie algebras and their representations. Monographs and textbooks in Pure and Applied math, 116. 1988.
  • P. D. Francesco, O. Golinelli, and E. Guitter. Meanders and the Temperley-Lieb algebra. Comm. Math. Phys., 186:1–59, 1997.
  • I. B. Frenkel, N. Jing, and W. Wang. Vertex representations via finite groups and the Mckay correspondence. arXiv:math.QA/9907166
  • I. B. Frenkel and M. Khovanov. Canonical bases in tensor products and graphical calculus for ${U}_q(\frak{sl}_2)$. Duke Math J., 87(3):409–480, 1997.
  • K. Fukaya. Floer homology for 3-manifolds with boundary. In Topology geometry and field theory, pages 1–21. World Sci. Publishing, River Edge, NJ, 1994.
  • K. Fukaya and P. Seidel. Floer homology, ${A}_{\infty}$-categories and topological field theory. In Geometry and physics (Aarhus, 1995), Lecture notes in Pure and Appl. Math., pages 9–32. Dekker, New York, 1997.
  • D. Happel. Triangulated categories in the representation theory of finite-dimensional algebras. London Math. Soc. Lect. Note Ser. 119. Cambridge University Press, 1988.
  • R. S. Huerfano and M. Khovanov. A category for the adjoint representation. arXiv:math.QA/0002060.
  • B. Iversen. Cohomology of sheaves. Springer-Verlag, 1987.
  • J. C. Jantzen. Representations of algebraic groups. Pure and Appl. Math. 131. Academic Press, Inc., Boston, 1987.
  • V. F. R. Jones. A polynomial invariant for knots via von Newmann algebras. Bull. AMS, 12(1):103–111, 1985.
  • L. Kadison. New examples of Frobenius extensions. University lecture series 14. AMS, 1999.
  • M. Kapranov and E. Vasserot. Kleinian singularities, derived categories, and Hall algebras. arXiv:math.AG/9812016
  • M. Kashiwara and P. Schapira. Sheaves on manifolds. Grundlehren der mathematischen Wissenschaften 292. Springer-Verlag, 1990.
  • L. H. Kauffman. State models and the Jones polynomial. Topology, 26(3):395–407, 1987.
  • L. H. Kauffman and S. L. Lins. Temperley-Lieb recoupling theory and invariants of 3-manifolds. Annals of Math. Studies, 134. Princeton University Press, 1994.
  • M. Khovanov. NilCoxeter algebras categorify the Weyl algebra. To appear in Communications in Algebra. arXiv:math.RT/9906166
  • M. Khovanov. Graphical calculus, canonical bases and Kazhdan-Lusztig theory. PhD thesis, Yale University, 1997.
  • M. Khovanov. A categorification of the Jones polynomial. Duke Math J., 101(3):359–426, 1999. arXiv:math.QA/9908171
  • M. Khovanov and P. Seidel. Quivers, Floer homology, and braid group actions. arXiv:math.QA/0006056
  • A. N. Kirillov and N. Y. Reshetikhin. Representations of the algebra ${U}_q(sl(2)),$ q-orthogonal polynomials and invariants of links. In V.G.Kac, editor, Infinite dimensional Lie algebras and groups. World Scientific, 1989.
  • M. Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the International Congress of Mathematicians, (Zürich, 1994), pages 120–139. Birkhäuser, 1995. arXiv:math.AG/9411018
  • G. Kuperberg. Spiders for rank $2$ Lie algebras. Comm. Math. Phys., 180(1):109–151, 1996. arXiv:math.QA/9712003
  • T. Y. Lam. Lectures on modules and rings. Graduate texts in mathematics 189. Springer-Verlag, 1999.
  • M. Mackaay. Finite groups, spherical 2-categories, and 4-manifold invariants. arXiv:math.QA/9903003
  • G. Malle and A. Mathas. Symmetric cyclotomic Hecke algebras. Journal of Algebra, 205:275–293, 1998.
  • H. Nakajima. Instantons on ALE spaces, quiver varieties and Kac-Moody algebras. Duke Mathematical Journal, 76:365–416, 1994.
  • A. Polishchuk. Gluing of perverse sheaves on the basic affine space. arXiv:math.AG/9811155
  • N. Reshetikhin and V. Turaev. Invariants of $3$-manifolds via link polynomials and quantum groups. Invent. Math., 103(3):547–597, 1991.
  • J. Rickard. Triangulated categories in the modular representation theory of finite groups. In Derived equivalences for group rings, volume 1685 of Lecture Notes in Math., pages 177–198. Springer-Verlag, 1998.
  • R. Rouquier. Action du groupe de tresses sur la catégorie dériveé de la variété de drapeaux. preprint,\char'176rouquier/
  • R. Rouquier. Travaux de recherches. Représentations et catégories dérivées.\char'176rouquier/, 1998.
  • R. Rouquier and A. Zimmermann. Picard groups for derived module categories. preprint,\char'176rouquier/, 1998.
  • P. Seidel and R. Thomas. Braid group actions on derived categories of coherent sheaves. Duke Mathematical Journal, to appear, 2000. arXiv:math.AG/0001043
  • M. E. Sweedler. Hopf algebras. W. A. Benjamin, Inc., New York, 1969.
  • V. Turaev. Quantum invariants on knots and 3-manifolds. de Gruiter studies in mathematics, 18. 1994.
  • W. Wang. Hilbert schemes, wreath products, and the Mckay correspondence. arXiv:math.AG/9912104
  • E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys., 121(3):351–399, 1989.