The Kauffman polynomial of links and representation theory. (English) Zbl 0666.57006

The author constructs algebras from knot invariants in a reversal of the construction of one of the recent knot polynomials from the Iwahori (Hecke) algebras. Here the Kauffman polynomial is used. A geometric construction produces a sequence of algebras \(E_ n\), based on \(n\)-tangles (here called ‘knits’), and an evaluation map on \(E_ n\), which calculates the Kauffman regular isotopy polynomial for the closure of each \(n\)-tangle represented in \(E_ n\).
In the case \(n=3\) he describes another algebra \(\overline E_ 3\) by generators and relations, isomorphic to \(E_ 3\). He goes on to study the structure and representations of \(E_ 3\), and deduces results about the Kauffman polynomial for 2-bridge links and closed 3-braids. Independently, J. S. Birman and H. Wenzl [Braids, link polynomials and a new algebra (Preprint, Columbia Univ. 1986), Trans. Am. Math. Soc. 313, No. 1, 249–273 (1989; Zbl 0684.57004)], with similar motivation, have given a sequence of algebras in terms of generators and relations. For \(n=3\) their algebra agrees with the author’s algebra \(\overline E_ 3\).
More recent work has shown that the algebraically and geometrically defined algebras can be directly identified for each \(n\).


57M25 Knots and links in the \(3\)-sphere (MSC2010)


Zbl 0684.57004