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Knots, groups, subfactors and physics

  • Takagi Lectures
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Japanese Journal of Mathematics Aims and scope

Abstract

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.

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References

  1. Baxter R.J.: Exactly solved models in statistical mechanics. Academic Press, London (1982)

    MATH  Google Scholar 

  2. Cannon J.W., Floyd W.J., Parry W.R.: Introductory notes on Richard Thompson’s groups. Enseign. Math. 42, 215–256 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Connes A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. (4) 6, 133–252 (1973)

    MathSciNet  MATH  Google Scholar 

  4. Connes A.: Sur la classification des facteurs de type II. C. R. Acad. Sci. Paris Sér. A-B 281, 13–15 (1975)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  7. A. Connes, Noncommutative Geometry, Academic Press, 1994.

  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.

  9. D.E. Evans and Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Math. Monogr., Oxford Univ. Press, 1998.

  10. G. Golan and M. Sapir, On Jones’ subgroup of R. Thompson group \({F}\), preprint, arXiv:1501.00724.

  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.

  12. Gordon C.McA., Luecke J.: Knots are determined by their complements. J. Amer. Math. Soc. 2, 371–415 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  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.

  15. Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jones V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Amer. Math. Soc. (N.S.) 12, 103–111 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. V.F.R. Jones, Planar algebras I, preprint, arXiv:math/9909027.

  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.

  19. V.F.R. Jones, Some unitary representations of Thompson’s groups F and T, preprint, arXiv:1412.7740.

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kauffman L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  22. McDuff D.: Uncountably many \({{\mathrm{II}}_1}\) factors. Ann. of Math. (2) 90, 372–377 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  23. Murray F.J., von Neumann J.: On rings of operators. Ann. of Math. (2) 37, 116–229 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  24. Murray F.J., von Neumann J.: On rings of operators. IV. Ann. of Math. (2) 44, 716–808 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nakamura M., Takeda Z.: A Galois theory for finite factors. Proc. Japan Acad. 36, 258–260 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  27. Popa S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120, 427–445 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Powers R.T.: Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. of Math. (2) 86, 138–171 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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.

  30. D.-V. Voiculescu, K.J. Dykema and A. Nica, Free Random Variables, CRM Monogr. Ser., 1, Amer. Math. Soc., Providence, RI, 1993.

  31. von Neumann J.: On infinite direct products. Compositio Math. 6, 1–77 (1939)

    MathSciNet  MATH  Google Scholar 

  32. von Neumann J.: On rings of operators. Reduction theory. Ann. of Math. (2) 50, 401–485 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wenzl H.: On sequences of projections. C. R. Math. Rep. Acad. Canada 9, 5–9 (1987)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1362 Stevenson Center, Nashville, TN, 37240, USA

    Vaughan F. R. Jones

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Vaughan F. R. Jones

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  1. Vaughan F. R. Jones
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Correspondence to Vaughan F. R. Jones.

Additional information

Communicated by: Yasuyuki Kawahigashi

This article is based on the 15th Takagi Lectures that the author delivered at Tohoku University on June 27 and 28, 2015.

Vaughan Jones is supported by the NSF under Grant No. DMS-0301173.

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Jones, V.F.R. Knots, groups, subfactors and physics. Jpn. J. Math. 11, 69–111 (2016). https://doi.org/10.1007/s11537-016-1529-x

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  • DOI: https://doi.org/10.1007/s11537-016-1529-x

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