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The Yang-Baxter equation and invariants of links. (English) Zbl 0648.57003
The Yang-Baxter operators which arise in the theory of exactly solved models of statistical mechanics satisfy equations analogous to the classical braid relations. In this paper it is shown how an “extended YB operator” S on the tensor powers of a finitely generated free module over a commutative ring K gives rise to a map from the set of all braids to K which takes the same value on Markov-equivalent braids and so defines a K-valued invariant \(T_ S\) of oriented links. If the homomorphism underlying S satisfies a polynomial equation then \(T_ S\) satisfies a corresponding generalized Conway identity. Many examples may be constructed via representations of simple Lie algebras. In particular the “homfly” and Kauffman 2-variable polynomials arise in this way. Under some additional assumptions on S, the invariant \(T_ S\) can be given a state model description. The questions as to how these invariants may be understood in terms of algebraic topology and whether they may be extended to links in other 3-manifolds remain open.
Reviewer: J.Hillman

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
20F36 Braid groups; Artin groups
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References:
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