×

zbMATH — the first resource for mathematics

Braids, link polynomials and a new algebra. (English) Zbl 0684.57004
This paper uses Kauffman’s construction of his 2-variable link polynomial as a guide to construct a new 2-parameter family of semisimple, finite- dimensional algebras with trace functions which, when normalized, give back the Kauffman polynomial. After a review of the Hecke algebra approach to the homfly polynomial, the new algebras \({\mathcal C}_ n(\ell,m)\) are introduced via generators and relations, and the Kauffman polynomial is used to define trace functions. The “basic construction” technique of Jones is used to investigate the structure of these algebras. It is also shown that they may be regarded as deformations of the “centralizer algebras” constructed by R. Brauer in 1937. The final section of the paper considers an application to the question of whether the braid groups are linear groups, and asks whether a particular 6- dimensional representation of \(B_ 4\) is faithful.
Reviewer: J.Hillman

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F29 Representations of groups as automorphism groups of algebraic systems
20F36 Braid groups; Artin groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. W. Alexander, A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. U.S.A. 9 (1923), 93-95.
[2] E. Artin, Theorie der Zopfe, Hamburg Abh. 4 (1925), 47-72. · JFM 51.0450.01
[3] Joan S. Birman, Braids, links, and mapping class groups, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. · Zbl 0297.57001
[4] 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. · Zbl 0595.57009 · doi:10.1007/BF01388747 · doi.org
[5] Claude Bourin, Unpublished notes.
[6] Richard Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857 – 872. · Zbl 0017.39105 · doi:10.2307/1968843 · doi.org
[7] 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. · Zbl 0572.57002
[8] Jim Hoste, A polynomial invariant of knots and links, Pacific J. Math. 124 (1986), no. 2, 295 – 320. · Zbl 0614.57005
[9] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1 – 25. · Zbl 0508.46040 · doi:10.1007/BF01389127 · doi.org
[10] -, Braid groups, Hecke algebras and type II factors, Geometric Methods in Abstract Algebras, Proc. U.S.-Japan Symposium, Wiley, 1986, pp. 242-273.
[11] 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. · Zbl 0564.57006
[12] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), no. 2, 335 – 388. · Zbl 0631.57005 · doi:10.2307/1971403 · doi.org
[13] Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417 – 471. · Zbl 0763.57004
[14] -, A states model for the Jones polynomial, Topology 26 (1987), 395-407. · Zbl 0622.57004
[15] -, Sign and space: knots and physics, Lecture Notes, Torino.
[16] M. Kidwell, On the dimension of the Birman-Wenzl algebra, Unpublished notes.
[17] Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. · Zbl 0193.34701
[18] W. B. R. Lickorish and Kenneth C. Millett, A polynomial invariant of oriented links, Topology 26 (1987), no. 1, 107 – 141. · Zbl 0608.57009 · doi:10.1016/0040-9383(87)90025-5 · doi.org
[19] W. B. R. Lickorish, A relationship between link polynomials, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 1, 109 – 112. · Zbl 0603.57005 · doi:10.1017/S0305004100065890 · doi.org
[20] George Lusztig, On a theorem of Benson and Curtis, J. Algebra 71 (1981), no. 2, 490 – 498. · Zbl 0465.20042 · doi:10.1016/0021-8693(81)90188-5 · doi.org
[21] Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745 – 758. · Zbl 0666.57006
[22] H. Morton and H. B. Short, Calculating the \( 2\)-variable polynomial for knots presented as closed braids, preprint. · Zbl 0738.57003
[23] H. Morton and P. Traczyk, Knots, skeins and algebras, preprint. · Zbl 1290.57021
[24] A. Ocneanu, A polynomial invariant for knots; a combinatorial and algebraic approach, preprint. · Zbl 0487.46037
[25] 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. · Zbl 0838.05103 · doi:10.1090/conm/178/01904 · doi.org
[26] Hans Wenzl, Hecke algebras of type \?_\? and subfactors, Invent. Math. 92 (1988), no. 2, 349 – 383. · Zbl 0663.46055 · doi:10.1007/BF01404457 · doi.org
[27] -, On the structure of Brauer’s centralizer algebras, Ann. of Math. 128 (1988), 179-193. · Zbl 0656.20040
[28] David Yetter, Private correspondence with the first author. · Zbl 0873.57016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.