Japanese Journal of Mathematics

, Volume 11, Issue 1, pp 69–111 | Cite as

Knots, groups, subfactors and physics

Takagi Lectures


Groups have played a big role in knot theory. We show how subfactors (subalgebras of certain von Neumann algebras) lead to unitary representations of the braid groups and Thompson’s groups \({F}\) and \({T}\). All knots and links may be obtained from geometric constructions from these groups. And invariants of knots may be obtained as coefficients of these representations. We include an extended introduction to von Neumann algebras and subfactors.

Keywords and phrases

subfactors planar algebras 

Mathematics Subject Classification (2010)

46L37 (46L54) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baxter R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1982)MATHGoogle Scholar
  2. 2.
    Cannon J.W., Floyd W.J., Parry W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math. 42, 215–256 (1996)MathSciNetMATHGoogle Scholar
  3. 3.
    Connes A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. (4) 6, 133–252 (1973)MathSciNetMATHGoogle Scholar
  4. 4.
    Connes A.: Sur la classification des facteurs de type II. C. R. Acad. Sci. Paris Sér. A-B 281, 13–15 (1975)MathSciNetMATHGoogle Scholar
  5. 5.
    Connes A.: Classification of injective factors. Cases \({{\mathrm{II}}_1}\), \({{\mathrm{II}}_\infty}\), \({{\mathrm{III}}_\lambda}\), \({\lambda \neq 1}\). Ann. of Math. (2) 104, 73–115 (1976)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    A. Connes, Sur la théorie non commutative de l’intégration, In: Algèbres d’opérateurs, Lecture Notes in Math., 725, Springer-Verlag, 1979, pp. 19–143.Google Scholar
  7. 7.
    A. Connes, Noncommutative Geometry, Academic Press, 1994.Google Scholar
  8. 8.
    J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, In: Computational Problems in Abstract Algebra, Proc. Conf., Oxford, 1967, Pergamon, Oxford, 1970, pp. 329–358.Google Scholar
  9. 9.
    D.E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Math. Monogr., Oxford Univ. Press, 1998.Google Scholar
  10. 10.
    G. Golan and M. Sapir, On Jones’ subgroup of R. Thompson group \({F}\), preprint, arXiv:1501.00724.
  11. 11.
    F.M. Goodman, P. de la Harpe and V.F.R. Jones, Coxeter Graphs and Towers of Algebras, Math. Sci. Res. Inst. Publ., 14, Springer-Verlag, 1989.Google Scholar
  12. 12.
    Gordon C.McA., Luecke J.: Knots are determined by their complements. J. Amer. Math. Soc. 2, 371–415 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    A. Guionnet, V.F.R. Jones and D. Shlyakhtenko, Random matrices, free probability, planar algebras and subfactors, In: Quanta of Maths, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010, pp. 201–239.Google Scholar
  14. 14.
    U. Haagerup, Principal graphs of subfactors in the index range \({4 < [M:N] < 3+\sqrt2}\), In: Subfactors, (eds. H. Araki, Y. Kawahigashi and H. Kosaki), World Sci. Publ., River Edge, NJ, 1994, pp. 1–38.Google Scholar
  15. 15.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jones V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.) 12, 103–111 (1985)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    V.F.R. Jones, Planar algebras I, preprint, arXiv:math/9909027.
  18. 18.
    V.F.R. Jones, The annular structure of subfactors, In: Essays on Geometry and Related Topics, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001, pp. 401–463.Google Scholar
  19. 19.
    V.F.R. Jones, Some unitary representations of Thompson’s groups F and T, preprint, arXiv:1412.7740.
  20. 20.
    Jones V.F.R., Morrison S., Snyder N.: The classification of subfactors of index at most 5. Bull. Amer. Math. Soc. (N.S.) 51, 277–327 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kauffman L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    McDuff D.: Uncountably many \({{\mathrm{II}}_1}\) factors. Ann. of Math. (2) 90, 372–377 (1969)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Murray F.J., von Neumann J.: On rings of operators. Ann. of Math. (2) 37, 116–229 (1936)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Murray F.J., von Neumann J.: On rings of operators. IV. Ann. of Math. (2) 44, 716–808 (1943)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nakamura M., Takeda Z.: A Galois theory for finite factors. Proc. Japan Acad. 36, 258–260 (1960)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, In: Operator Algebras and Applications. Vol. 2, London Math. Soc. Lecture Note Ser., 136, Cambridge Univ. Press, 1988, pp. 119–172.Google Scholar
  27. 27.
    Popa S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120, 427–445 (1995)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Powers R.T.: Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. of Math. (2) 86, 138–171 (1967)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    H.N.V. Temperley and E.H. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser. A, 322 (1971), 251–280.Google Scholar
  30. 30.
    D.-V. Voiculescu, K.J. Dykema and A. Nica, Free Random Variables, CRM Monogr. Ser., 1, Amer. Math. Soc., Providence, RI, 1993.Google Scholar
  31. 31.
    von Neumann J.: On infinite direct products. Compositio Math. 6, 1–77 (1939)MathSciNetMATHGoogle Scholar
  32. 32.
    von Neumann J.: On rings of operators. Reduction theory. Ann. of Math. (2) 50, 401–485 (1949)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Wassermann A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of \({LSU(N)}\) using bounded operators. Invent. Math. 133, 467–538 (1998)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wenzl H.: On sequences of projections. C. R. Math. Rep. Acad. Canada 9, 5–9 (1987)MathSciNetMATHGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2016

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA
  2. 2.Department of MathematicsKyoto UniversityKyotoJapan

Personalised recommendations