The Mathematical Intelligencer

, Volume 13, Issue 1, pp 52–60 | Cite as

Recent developments in braid and link theory

  • Joan S. Birman


Steklov Institute Braid Group Link Type Jones Polynomial Link Diagram 
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  1. 1.
    J. W. Alexander, A lemma on systems of knotted curves,Proc. Nat. Acad. Sciences USA 9 (1923), 93–95.CrossRefGoogle Scholar
  2. 2.
    E. Artin, Theorie der Zöpfe,Hamburg. Abh. 4 (1925), 47–72.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    R. J. Baxter,Exactly Solved Models in Statistical Mechanics. London: Academic Press (1982).zbMATHGoogle Scholar
  4. 4.
    D. Bennequin, Entrelacements et equations de Pfaff,As-térisque 107–108 (1983), 87–161.MathSciNetGoogle Scholar
  5. 5.
    J. S. Birman, Braids, links and mapping class groups.Ann. of Math. Studies No. 82, Princeton Univ. Press (1974).Google Scholar
  6. 6.
    J. Birman and W. Menasco, Closed braid representatives of the unlink, preprint, 1989.Google Scholar
  7. 7.
    ---, On the classification of links that are closed 3-braids, preprint (1989).Google Scholar
  8. 8.
    J. S. Birman and H. Wenzl, Braids, link polynomials and a new algebra.Trans. AMS, to appear, preprint, New York: Columbia Univ.Google Scholar
  9. 9.
    G. Burde and H. Zieschgang, Knots. Berlin: de Gruyter (1986).Google Scholar
  10. 10.
    L. Crane, Topology of 3-manifolds and conformai field theories, preprint, Yale Univ. (1989).Google Scholar
  11. 11.
    E. Fadell and L. Neuwirth, Configuration spaces,Math. Scand. 10 (1962), 111–118.zbMATHMathSciNetGoogle Scholar
  12. 12.
    P. Freyd, J. Hoste, W. Lickorish, K. Millett, A. Ocneanu and D. Yetter, A new polynomial invariant of knots and links,Bull Amer. Math. Soc. (2)12 (1985), 257–267.CrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Jimbo, Quantum R-matrix related to the generalized Toda system: an algebraic approach,Lecture Notes in Physics 246 (1986), 335–361.CrossRefMathSciNetGoogle Scholar
  14. 14.
    V. Jones, Braid groups, Hecke algebras and type IIj factors,Proc. US Japan Seminar Kyoto (Araki and Effros, eds.) New York: John Wiley (1973).Google Scholar
  15. 15.
    —, Hecke algebra representation of braid groups and link polynomials,Ann. of Math. (2)126 (1987), 335–388.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    L. Kauffman, States models and the Jones polynomial,Topology 26 (1987), 395–407.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    ---, An invariant of regular isotopy,Trans. Amer. Math. Soc, to appear.Google Scholar
  18. 18.
    T. Kohno, Linear representations of braid groups and classical Yang-Baxter equations,BRAIDS, Contemp. Math. 78, 339–364,Amer. Math. Soc. (1988).CrossRefMathSciNetGoogle Scholar
  19. 19.
    A. King and M. Rocek, The Burau representation and the Alexander polynomial, preprint, Stony Brook: SUNY (1988).Google Scholar
  20. 20.
    H. Morton, Threading knot diagrams,Math Proc. Camb. Phil. Soc. 99 (1986), 246–260.Google Scholar
  21. 21.
    J. Przytycki and P. Traczyk, Invariants of links of Conway type,Kobe J. Math. 4 (1987), 115–139.zbMATHMathSciNetGoogle Scholar
  22. 22.
    K. Reidemeister,Knotentheorie. New York: Chelsea Pub. Co. (1948). English translation:Knot Theory, BSC Associates, Moscow: ID (1983).Google Scholar
  23. 23.
    N. Reshetiken, Quantized universal enveloping algebras, the Yang-Baxter equation, and invariants of links I and II, preprint, Leningrad: Steklov Institute of Math. (1987).Google Scholar
  24. 24.
    D. Rolfsen,Knots and Links, Berkeley: Publish or Perish (1976).zbMATHGoogle Scholar
  25. 25.
    P. G. Tait, On Knots I, II, III.Scientific papers I, London: Camb. Univ Press (1898).Google Scholar
  26. 26.
    V. Turaev, The Yang-Baxter equation and invariants of links, preprint, Leningrad: Steklov Institute of Math (1987).Google Scholar
  27. 27.
    E. Witten, Quantum field theory and the Jones polynomial, preprint, Institute for Advanced Study (1988).Google Scholar
  28. 28.
    S. Yamada. The minimum number of Seifert circles equals the braid index of a link,Invent. Math. 89 (1987), 347–356.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 1991

Authors and Affiliations

  • Joan S. Birman
    • 1
  1. 1.Department of MathematicsColumbia University and Barnard CollegeNew YorkUSA

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