Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Braids, link polynomials and a new algebra

Authors: Joan S. Birman and Hans Wenzl
Journal: Trans. Amer. Math. Soc. 313 (1989), 249-273
MSC: Primary 57M25; Secondary 17B99, 20F36
MathSciNet review: 992598
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on $ 2$ parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras.

References [Enhancements On Off] (What's this?)

  • [Al] J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93-95.
  • [Ar] E. Artin, Theorie der Zopfe, Hamburg Abh. 4 (1925), 47-72.
  • [Bi] J. S. Birman, Braids, links and mapping class groups, Ann. of Math. Studies, no. 82, Princeton Univ. Press, 1974. MR 0375281 (51:11477)
  • [B-L-M] R. Brandt, W. B. R. Lickorish and K. Millett, A polynomial invariant for unoriented knots and links, preprint. MR 837528 (87m:57003)
  • [Bo] Claude Bourin, Unpublished notes.
  • [Br] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872. MR 1503378
  • [FYHLMO] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 239-246. MR 776477 (86e:57007)
  • [H] J. Hoste, A polynomial invariant of knots and links, Pacific J. Math. 124 (1986), 295-320. MR 856165 (88d:57004)
  • [J,1] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. MR 696688 (84d:46097)
  • [J,2] -, Braid groups, Hecke algebras and type II factors, Geometric Methods in Abstract Algebras, Proc. U.S.-Japan Symposium, Wiley, 1986, pp. 242-273.
  • [J,3] -, A polynomial invariant for knots via Von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), 103-111. MR 766964 (86e:57006)
  • [J,4] -, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (1987), 335-388. MR 908150 (89c:46092)
  • [K,1] L. Kauffman, An invariant of regular isotopy, preprint. MR 958895 (90g:57007)
  • [K,2] -, A states model for the Jones polynomial, Topology 26 (1987), 395-407.
  • [K,3] -, Sign and space: knots and physics, Lecture Notes, Torino.
  • [Ki] M. Kidwell, On the dimension of the Birman-Wenzl algebra, Unpublished notes.
  • [La] S. Lang, Algebra, Addison-Wesley, 1965. MR 0197234 (33:5416)
  • [L-M] W. B. R. Lickorish and K. Millett, A polynomial invariant of oriented links, Topology 26 (1987), 107-141. MR 880512 (88b:57012)
  • [Li] W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Philos. Soc. 100 (1986), 109-112. MR 838656 (87g:57010)
  • [Lu] George Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981). MR 630610 (83a:20053)
  • [M] J. Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745-758. MR 927059 (89c:57007)
  • [M-S] H. Morton and H. B. Short, Calculating the $ 2$-variable polynomial for knots presented as closed braids, preprint.
  • [M-T] H. Morton and P. Traczyk, Knots, skeins and algebras, preprint.
  • [O] A. Ocneanu, A polynomial invariant for knots; a combinatorial and algebraic approach, preprint.
  • [S] S. Sundaram, On the combinatorics of representations of $ {S_p}(2n;{\mathbf{C}})$, Mem. Amer. Math. Soc. (to appear). MR 1310588 (96f:05193)
  • [W,1] H. Wenzl, Representations of Hecke algebras and subfactors, Invent. Math. 24 (1988), 349-383. MR 936086 (90b:46118)
  • [W,2] -, On the structure of Brauer's centralizer algebras, Ann. of Math. 128 (1988), 179-193.
  • [Y] David Yetter, Private correspondence with the first author.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57M25, 17B99, 20F36

Retrieve articles in all journals with MSC: 57M25, 17B99, 20F36

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society