Advertisement

An Introduction to Knot Theory and Functional Integrals

  • L. Kauffman
Part of the NATO ASI Series book series (NSSB, volume 361)

Abstract

This paper is an expanded version of a course of lectures on knots and functional integration delivered in Corsica during September 1996.

Keywords

Wilson Loop Wilson Line Jones Polynomial Reidemeister Move Chord Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L. H. Kauffman, Functional Integration and the theory of knots, J. Math. Physics, Vol. 36(5), May 1995, pp. 2402–2429.MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. [2]
    K. Reidemeister, Knotentheorie, Chelsea Pub. Co., New York, 1948, Copyright 1932, Julius Springer, Berlin.Google Scholar
  3. [3]
    H. Trotter, Non-invertible knots exist, Topology, Vol. 2, 1964, pp. 275–280.MathSciNetCrossRefGoogle Scholar
  4. [4]
    J. H. White, Self-linking and the Gauss integral in higher dimensions. Amer. J. Math., 91 (1969), pp. 693–728.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D. Joyce, A classifying invariant for knots:the knot quandle, J. Pure and Appl. Alg, Vol. 23, 1982, pp. 37–65.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    L. H. Kauffman, Knots and Physics, World Scientific Pub., 1991 and 1993.Google Scholar
  7. [7]
    R. Fenn and C. Rourke, Racks and links in codimension two, J. of Knot Theory and Its Ramif., Vol. 1, No. 4, 1992, pp. 343–406.MathSciNetCrossRefGoogle Scholar
  8. [8]
    L. H. Kauffman and S. K. Winker, Quandles, Crystals and Racks — A New Approach to Knot Theory, World Scientific Pub., 1997 (to appear).Google Scholar
  9. [9]
    J. H. Conway, An enumeration of knots and links and some of their algebraic properties, in Computational Problems in Abstract Algebra, Pergammon Press, N. Y., 1970, pp. 329–358.Google Scholar
  10. [10]
    V. F. R. Jones, A polynomial invariant for links via von Neumann algebras, Bull. Amer. Math. Soc., Vol. 129, 1985, pp. 103–112.CrossRefGoogle Scholar
  11. [11]
    R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Acad. Press, 1982.Google Scholar
  12. [12]
    V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math., Vol. 126, 1987, pp. 335–388.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc., Vol. 318. No. 2, 1990, pp. 417–471.MathSciNetCrossRefGoogle Scholar
  14. [14]
    L. H. Kauffman, State Models and the Jones Polynomial, Topology, Vol. 26, 1987, pp. 395–407.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    L. H. Kauffman, Statistical mechanics and the Jones polynomial, AMS Contemp. Math. Series, Vol. 78, 1989, pp. 263–297.MathSciNetCrossRefGoogle Scholar
  16. [16]
    V. F. R. Jones, On knot invariants related to some statistical mechanics models, Pacific J. Math., Vol. 137, No. 2, 1989, pp. 311–334.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    V. G. Turaev, The Yang-Baxter equations and invariants of links, LOMI preprint E-3-87, Steklov Institute, Leningrad, USSR., Inventiones Math., Vol. 92, Fasc.3, pp. 527-553.Google Scholar
  18. [18]
    L. H. Kauffman, Gauss codes, quantum groups and ribbon Hopf algebras, Reviews in Mathematical Physics, Vol. 5, No. 4., 1993, pp. 735–773.MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    N. Y. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I and II, LOMI reprints E-4-87 and E-17-87, Stekiov Institute, Leningrad, USSR.Google Scholar
  20. [20]
    L. H. Kauffman, The Conway polynomial, Topology, 20 (1980), pp. 101–108.MathSciNetCrossRefGoogle Scholar
  21. [21]
    L. H. Kauffman, On Knots, Annals Study No. 115, Princeton University Press (1987)Google Scholar
  22. [22]
    L. H. Kauffman and D. Radford, Invariants of 3-Manifolds derived from finite dimensional Hopf algebras, Journal of Knot Theory and Its Ramifications. Vol. 4, No. 1 (1995), 131–162.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    L. H. Kauffman, Hopf algebras and 3-Manifold invariants, Journal of Pure and Applied Algebra. Vol. 100 (1995), 73–92.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    D. Yetter, Quantum groups and representations on monoidal categories. Math. Proc. Camb. Phil. Soc., Vol. 108 (1990), pp. 197–229.MathSciNetCrossRefGoogle Scholar
  25. [25]
    M. A. Hennings, Hopf algebras and regular isotopy invariants for link diagrams, Math. Proc. Camb. Phil. Soc., Vol. 109 (1991), pp. 59–77MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Lee Smolin, Link polynomials and critical points of the Chern-Simons path integrals, Mod. Phys. Lett. A, Vol. 4, No. 12, 1989, pp. 1091–1112.MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys., Vol. 121, 1989, pp. 351–399.MathSciNetADSzbMATHCrossRefGoogle Scholar
  28. [28]
    M. F. Atiyah, Geometry of Yang-Mills Fields, Accademia Nazionale dei Lincei Scuola Superiore Lezioni Fermiare, Pisa, 1979.Google Scholar
  29. [29]
    N. Y. Reshetikhin and V. Turaev, Invariants of Three Manifolds via link polynomials and quantum groups, Invent. Math., Vol. 103, 1991, pp. 547–597.MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [30]
    V. G. Turaev and H. Wenzl, Quantum invariants of 3-manifolds associated with classical simple Lie algebras, International J. of Math., Vol. 4, No. 2, 1993, pp. 323–358.MathSciNetCrossRefGoogle Scholar
  31. [31]
    R. Kirby and P. Melvin, On the 3-manifold invariants of Reshetikhin-Turaev for sl (2, C), Invent. Math. 105, 473–545, 1991.MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. [32]
    W. B. R. Lickorish, The Temperley Lieb Algebra and 3-manifold invariants, Journal of Knot Theory and Its Ramifications, Vol. 2, 1993, pp. 171–194.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    K. Walker, On Witten’s 3-Manifold Invariants, (preprint 1991).Google Scholar
  34. [34]
    L. H. Kauffman and S. Lins, Temperley Lieb Recoupling Theory and Invariants of 3-Manifolds, Annals of Mathematics Studies Number 134, Princeton University Press, 1994.Google Scholar
  35. [35]
    M. F. Atiyah, The Geometry and Physics of Knots, Cambridge University Press, 1990.Google Scholar
  36. [36]
    S. Garoufalidis, Applications of TQFT to invariants in low dimensional topology, (preprint 1993).Google Scholar
  37. [37]
    D. Freed and R. Gompf, Computer calculations of Witten’s 3-manifold invariants, Comm. Math. Phys., Vol. 41, pp. 79–117, 1991.MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    L. C. Jeffrey, On Some Aspects of Chern-Simons Gauge Theory, Thesis — Oxford, 1991.Google Scholar
  39. [39]
    L. Rozansky, A large k asymptotics of Witten’s invariant of Seifert manifolds, (preprint 1993).Google Scholar
  40. [40]
    P. Cotta-Ramusino, E. Guadagnini, M. Martellini, M. Mintchev, Quantum field theory and link invariants, (preprint 1990)Google Scholar
  41. [41]
    T. Cheng and L Li, Gauge Theory of Elementary Particle Physics, Clarendon Press, Oxford, 1988.Google Scholar
  42. [42]
    L. H. Kauffman, New invariants in the theory of knots, Amer. Math. Monthly, Vol. 95, No. 3, March 1988. pp. 195–242.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [43]
    L. H. Kauffman and P. Vogel, Link polynomials and a graphical calculus, Journal of Knot Theory and Its Ramifications, Vol. 1, No. 1, March 1992.Google Scholar
  44. [44]
    L. H. Kauffman, Formal Knot Theory, Princeton University Press, Lecture Notes Series #30 (1983).Google Scholar
  45. [45]
    V. Vassiliev, Cohomology of knot spaces, In Theory of Singularities and Its Applications, V. I. Arnold, ed., Amer. Math. Soc., 1990, pp. 23-69.Google Scholar
  46. [46]
    J. Birman and X. S. Lin, Knot polynomials and Vassiliev’s invariants, (to appear in Invent. Math.).Google Scholar
  47. [47]
    D. Bar-Natan, On the Vassiliev knot invariants, (to appear in Topology).Google Scholar
  48. [48]
    T. Stanford, Finite-type invariants of knots, links and graphs, (preprint 1992).Google Scholar
  49. [49]
    M. Kontsevich, Graphs, homotopical algebra and low dimensional topology, (preprint 1992).Google Scholar
  50. [50]
    T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Contemporary Mathematics, Vol. 78, Amer. Math. Soc., 1988, pp. 339–364.MathSciNetCrossRefGoogle Scholar
  51. [51]
    E. Guadagnini, M. Martellini and M. Mintchev, Chern-Simons model and new relations between the Homfly coefficients, Physics Letters B, Vol. 238, No. 4, Sept. 28 (1989), pp. 489–494.MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    Dror Bar-Natan, Perturbative Aspects of the Chern-Simons Topological Quantum field Theory, Ph. D. Thesis, Princeton University, June 1991.Google Scholar
  53. [53]
    Raoul Bott and Clifford Taubes, On the self-linking of knots, Jour. Math. Phys. 35 (1994), pp. 5247–5287.MathSciNetADSzbMATHCrossRefGoogle Scholar
  54. [54]
    Daniel Altshuler and Laurent Friedel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, preprint (1996).Google Scholar
  55. [55]
    J. Goldman and L. H. Kauffman, Rational Tangles, (to appear in Advances in Appl. Math.)Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • L. Kauffman
    • 1
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

Personalised recommendations