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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:
 [1] Baxter, E.J.: Exactly solved models in statistical mechanics. New York, London: Academic Press 1982 · Zbl 0538.60093 [2] Drinfel’d, V.G.: Quantum groups. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova155, 18-49 (1986) (in Russian) [3] Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc.12, 239-246 (1985) · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3 [4] Jimbo, M.: Quantum R matrix for the generalized Toda system. Commun. Math. Phys.102, 537-547 (1986) · Zbl 0604.58013 · doi:10.1007/BF01221646 [5] Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc.12, 103-111 (1985) · Zbl 0564.57006 · doi:10.1090/S0273-0979-1985-15304-2 [6] Jones, V.F.R.: Notes on a talk in Atiyah’s seminar. Handwritten manuscript (November 1986) [7] Kauffman, L.H.: New Invariants in the Theory of Knots. Preprint (1986) [8] Kulish, P.P., Sklyanin, E.K.: On the solutions of the Yang-Baxter equation. Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 95, 129-160 (1980) (in Russian) [9] Tahtadjan, L.A., Faddeev, L.D.: The quantum method for the inverse problem and the XYZ Heisenberg model. Uspekhi Mat. Nauk34, (N 5) 13-63 (1979); (English transl. Russian Math. Surv.34, (N 5) 11-61 (1979)) [10] Turaev, V.G.: The Reidemeister torsion in the knot theory. Uspekhi Mat. Nauk41, (N 1) 97-147 (1986); (English transl. Russian Math. Surv.41 (N 1) 119-182 (1986) · Zbl 0602.57005
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