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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)
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