Witten’s Approach, Braid Group Representations and X-Deformations

  • M. L. Ge
  • F. Piao
  • L. Y. Wang
  • K. Xue
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)


The Witten’s approach of link polynomials based on (2+1) Chern-Simons Lagrangian is used to simplify the calculations of braid group representations(BGR) for SU(2) algebra. On the basis of the direct derivations of BGR the “x-deformation” scheme is presented to generate explicitly the quantum R(x)-matrix for given BGR.


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  1. (1).
    Birman J.S. Ann Math Studies No.82,(1976) Princeton University Press.Google Scholar
  2. (2).
    Birman J.S. and Wenzel, H. “Braids,Link Polynomials and a new algebra”,to appear in Trans A.M.S.Google Scholar
  3. (3).
    Frohlich,J. “Statistics of fields,the Yang-Baxter equatoins and the theory of Knots and links,ETH-Honggerberg preprint.Google Scholar
  4. (4).
    Jones, V.F.R. Bull Amer Math Soc.12,103(1985) Ann of Math 126,335(1987)Google Scholar
  5. (5).
    ).Jones, V.F.R, V.F.R. “On knot invariants related to some statistical mechanics models ”,jpreprint 1988.Google Scholar
  6. (6).
    Kauffman L.H. “State model for knot polynomials— An introduction”,preprint. Ann of Math Studies # 115.Google Scholar
  7. (7).
    Kauffman, L.H, L.H. “Knot theory and Applications”,talk at UT Austin,March 1988Google Scholar
  8. (8).
    Kauffman, L.H. “Braid Group,Knot Theory and Statistical Mechanics” P.27,edited C.N.Yang,World Scientific,1989.Google Scholar
  9. (9).
    Akutsu,Y. and Wadati, M. J.Phys soc Jan,56, 839, 3039 (1987)CrossRefMathSciNetGoogle Scholar
  10. Akutsu, Y. and Wadati, M. “Braid Group,Knot Theory and Statistical Mechanics”,p 151, edited by C.N.Yang,World Scientific,1989.Google Scholar
  11. (10).
    Turaev, V.G. Invent Math 92, 527 (1988)MathSciNetGoogle Scholar
  12. (11).
    Reshetikhin, N. “Quantized universal enveloping algebras,the Yang-Baxter equation and invariants of links I,II” LOMT preprint E-4–87Google Scholar
  13. (12).
    kohno, T. “Quantum University enveloping algebras and monodromy of braid group” Nagoya preprint 1989Google Scholar
  14. kohno, T. Nankai lectures on Math Physics,Tianjin,1988.Google Scholar
  15. (13).
    Takhtajan, L., Nankai lectures on Quantum Group,Tianjin,1989.Google Scholar
  16. (14).
    Witten, E. Comm Math Phys, 121, 135 (1989)MathSciNetGoogle Scholar
  17. (15).
    Ge, M.L., Wang, L.Y., Xue, K., and Wu, Y.S., Inter J.Mod Phys 4, 3351 (1989).MathSciNetGoogle Scholar
  18. (16).
    Lee, H.C., Ge, M.L.,Couture, M. and Wu, Y.S., Inter J.Mod Phys 4, 2333 (1989)MathSciNetGoogle Scholar
  19. (17).
    Jimbo, M. Lett Math Phys 10, 63 (1985).MathSciNetGoogle Scholar
  20. (18).
    Jimbo, M. Comm Math Phys 102, 537 (1986).MathSciNetGoogle Scholar
  21. (19).
    Timbo, M., Lett Math Phys 11, 253 (1986)Google Scholar
  22. (20).
    Moore, G. and Seiberg, N., Phys Lett B212,451(1988),IAS preprint,HEP-88/39.Google Scholar
  23. (21).
    Kniznik, V G and Zamolodchikov, A.B. Nucl Phys B2147, 83 (1984).CrossRefADSGoogle Scholar
  24. (22).
    Ge, M.L., Wang L.Y. and Xue K, “Extended State Expandions and the University of Witten’ Version of Link Polynomial Theory” Nankai preprint 1989,to appear in Inter.Mod Phys.Google Scholar
  25. (23).
    Yamagishi, K., Ge M.L., and Wu Y.S., UU—HEP-89–1. “New Hierachies of Knot Polynomials from Topological Chern—Simons gauge Theory”.Google Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1990

Authors and Affiliations

  • M. L. Ge
    • 1
  • F. Piao
    • 1
  • L. Y. Wang
    • 1
  • K. Xue
    • 1
  1. 1.Theoretical Physics DivisionNankai Institute of MathematicsTianjinPeople’s Rep. of China

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