Braids, link polynomials and a new algebra

Birman, Joan S.; Wenzl, Hans

doi:10.1090/S0002-9947-1989-0992598-X

Braids, link polynomials and a new algebra
HTML articles powered by AMS MathViewer

by Joan S. Birman and Hans Wenzl

Trans. Amer. Math. Soc. 313 (1989), 249-273

DOI: https://doi.org/10.1090/S0002-9947-1989-0992598-X

PDF | Request permission

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

A lemma on systems of knotted curves

9

Theorie der Zopfe

4

Joan S. Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. MR 0375281
Robert D. Brandt, W. B. R. Lickorish, and Kenneth C. Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986), no. 3, 563–573. MR 837528, DOI 10.1007/BF01388747
Richard Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857–872. MR 1503378, DOI 10.2307/1968843
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), no. 2, 239–246. MR 776477, DOI 10.1090/S0273-0979-1985-15361-3
Jim Hoste, A polynomial invariant of knots and links, Pacific J. Math. 124 (1986), no. 2, 295–320. MR 856165
V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127

Braid groups, Hecke algebras and type

factors

Vaughan F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 103–111. MR 766964, DOI 10.1090/S0273-0979-1985-15304-2
V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335–388. MR 908150, DOI 10.2307/1971403
Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7

A states model for the Jones polynomial

26

Sign and space

knots and physics

On the dimension of the Birman-Wenzl algebra

Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107–141. MR 880512, DOI 10.1016/0040-9383(87)90025-5
W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 109–112. MR 838656, DOI 10.1017/S0305004100065890
George Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), no. 2, 490–498. MR 630610, DOI 10.1016/0021-8693(81)90188-5
Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059

Calculating the

variable polynomial for knots presented as closed braids

Knots, skeins and algebras

A polynomial invariant for knots; a combinatorial and algebraic approach

Sheila Sundaram, Applications of the Hopf trace formula to computing homology representations, Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 277–309. MR 1310588, DOI 10.1090/conm/178/01904
Hans Wenzl, Hecke algebras of type $A_n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349–383. MR 936086, DOI 10.1007/BF01404457

On the structure of Brauer’s centralizer algebras

128