Knot Invariants and Chern-Simons Theory

  • José M. F. Labastida
Conference paper
Part of the Progress in Mathematics book series (PM, volume 202)


A brief review of the development of Chern-Simons gauge theory since its relation to knot theory was discovered in 1988 is presented. The presentation is done guided by a dictionary which relates knot theory concepts to quantum field theory ones. From the basic objects in both contexts the quantities leading to knot and link invariants are introduced and analysed. The quantum field theory approaches that have been developed to compute these quantities are reviewed. Perturbative approaches lead to Vassiliev or finite type invariants. Non-perturbative ones lead to polynomial or quantum group invariants. In addition, a brief discussion on open problems and future developments is included.


Wilson Loop Double Point Feynman Rule Jones Polynomial Reidemeister Move 
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.


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  1. [1]
    D. Altschuler and L. Friedel, `Vassiliev knot invariants and Chern-Simons perturbation theory to all orders,“ Commun. Math. Phys. 187 (1997) 261, and `On universal Vassiliev invariants,” 170 (1995) 41.MathSciNetzbMATHGoogle Scholar
  2. [2]
    M. Alvarez and J. M. F. Labastida, “Analysis of observables in Chern-Simons perturbation theory,” Nucl. Phys. B395 (1993) 198, hep-th/9110069, and `Numerical knot invariants of finite type from Chern-Simons gauge theory,“ B433 (1995) 555, hep-th/9407076; Erratum, ibid. B441 (1995) 403.MathSciNetGoogle Scholar
  3. [3]
    M. Alvarez and J. M. F. Labastida, “Vassiliev invariants for torus knots,” Journal of Knot Theory and its Ramifications 5 (1996) 779; q-alg/9506009.Google Scholar
  4. [4]
    M. Alvarez and J. M. F. Labastida, “Primitive Vassiliev invariants and factorization in Chern-Simons gauge theory,” Commun. Math. Phys. 189 (1997) 641, qalg/9604010.Google Scholar
  5. [5]
    Y. Akutsu and M. Wadati, “Exactly solvable models and knot theory,” Phys. Rep. 180 (1989) 247.MathSciNetCrossRefGoogle Scholar
  6. [6]
    D. Bar-Natan “On the Vassiliev knot invariants,”Topology34 (1995) 423.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    D. Bar-Natan “Perturbative aspects of Chern-Simons topological quantum field theory”, Ph.D. Thesis, Princeton University, 1991.Google Scholar
  8. [8]
    J. S. Birman, “New points of view in knot theory,” Bull. AMS 28 (1993) 253.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J. S. Birman and X. S. Lin, “Knot polynomials and Vassiliev’s invariants,” Invent. Math. 111 (1993) 225.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    R. Bott and C. Taubes, “On the self-linking of knots,” Jour. Math. Phys. 35 (1994) 5247.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    A. S. Cattaneo, P. Cotta-Ramusino, J. Frohlich and M. Martellini, “Topological BF theories in three-dimensions and four-dimensions,” J. Math. Phys. 36 (1995) 6137.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millet and A. Ocneanu, “A new polynomial invariant of knots and links,” Bull. AMS 12 (1985) 239.zbMATHCrossRefGoogle Scholar
  13. [13]
    M. Goussarov, M. Polyak and O. Viro, “Finite Type Invariants of Classical and Virtual Knots”, preprint, 1998, math.GT/9810073.Google Scholar
  14. [14]
    E. Guadagnini, M. Martellini and M. Mintchev, `Perturbative aspects of the ChernSimons field theory,“ Phys. Lett. B227(1989) 111; ”Chern-Simons model and new relations between the HOMFLY coefficients,“ B228 (1989) 489, and ”Wilson lines in Chern-Simons theory and link invariants,“ Nucl. Phys. B330 (1990) 575.MathSciNetCrossRefGoogle Scholar
  15. [15]
    A. C. Hirshfeld and U. Sassenberg “Derivation of the total twist from Chern-Simons theory,”Journal of Knot Theory and its Ramifications5(1996) 489 and “Explicit formulation of a third order finite knot invariant derived from Chern-Simons theory,” 5(1996) 805MathSciNetzbMATHGoogle Scholar
  16. [16]
    V. F. R. Jones, “Hecke algebras representations of braid groups and link polynomials,” Ann. of Math. 126 (1987) 335.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    C. Kassel, M. Rosso and V. Turaev “Quantum groups and knot invariants”, Panoramas et syntheses 5, Societe Mathematique de France, 1997.Google Scholar
  18. [18]
    C. Kassel and V. Turaev, “Chord diagram invariants of tangles and graphs,” Duke Math. J. 92 (1998) 497–552.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    L. Kauffman, “Witten’s Integral and Kontsevich Integral”, Particles, Fields and Gravitation, Lodz, Poland 1998, Ed. Jakub Rembieliski; AIP Proceedings 453 (1998), 368.Google Scholar
  20. [20]
    L. H. Kauffman, “An invariant of regular isotopy,” Trans. Am. Math. Soc. 318 (1990) 417.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    M. Kontsevich, “Vassiliev’s knot invariants,” Advances in Soviet Math. 16, Part 2 (1993) 137.MathSciNetGoogle Scholar
  22. [22]
    J. M. F. Labastida, “Chern-Simons Gauge Theory: Ten Years After”, Trends in Theoretical Physics II, H. Falomir, R. Gamboa, F. Schaposnik, eds., American Institute of Physics, New York, 1999, CP 484, 1–41, hep-th/9905057.Google Scholar
  23. [23]
    J. M. F. Labastida and E. Pérez, “Kontsevich integral for Vassiliev invariants from Chern-Simons perturbation theory in the light-cone gauge,” J. Math. Phys. 39 (1998) 5183; hep-th/9710176.Google Scholar
  24. [24]
    J. M. F. Labastida and E. Pérez, “Gauge-invariant operators for singular knots in Chern-Simons gauge theory,” Nucl. Phys. B527 (1998) 499, hep-th/9712139.Google Scholar
  25. [25]
    J. M. F. Labastida and E. Pérez, “Combinatorial Formulae for Vassiliev Invariants from Chern-Simons Perturbation Theory”, J. Math. Phys. 41 (2000), 2658–2699.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    J. M. F. Labastida and E. Pérez, “Vassiliev Invariants in the Context of ChernSimons Gauge Theory”, Santiago de Compostela preprint, US-FT-18/98; hepth/9812105.Google Scholar
  27. [27]
    G. Leibbrandt, “Introduction to noncovariant gauges,” Rev. Mod. Phys. 59 (1987) 1067.MathSciNetCrossRefGoogle Scholar
  28. [28]
    M. Polyak and O. Viro, “Gauss diagram formulas for Vassiliev invariants,” Int. Math. Res. Notices 11 (1994) 445.MathSciNetCrossRefGoogle Scholar
  29. [29]
    H. Ooguri and C. Vafa, “Knot Invariants and Topological Strings”, Harvard preprint, HUTP-99/A070, hep-th/9912123.Google Scholar
  30. [30]
    D. Thurston, “Integral expressions for the Vassiliev knot Invariants”, Harvard University senior thesis, April 1995; math/9901110.Google Scholar
  31. [31]
    V. A. Vassiliev, “Cohomology of knot spaces”, Theory of singularities and its applications, Advances in Soviet Mathematics, vol. 1,Arvericam Math. Soc., Providence, RI, 1990, 23–69.Google Scholar
  32. [32]
    J. F. W. H. van de Wetering, “Knot invariants and universal R-matrices from perturbative Chern-Simons theory in the almost axial gauge,” Nucl. Phys. B379 (1992) 172.MathSciNetCrossRefGoogle Scholar
  33. [33]
    S. Willerton, “On Universal Vassiliev Invariants, Cabling, and Torus Knots”, University of Melbourne preprint (1998).Google Scholar
  34. [34]
    E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121 (1989) 351.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    S.-W. Yang, “Feynman integral, knot invariant and degree theory of maps”, National Taiwan University preprint, September 1997; q-alg/9709029.Google Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • José M. F. Labastida
    • 1
  1. 1.Departamento de Física de PartículasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain

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