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Knots, links, braids and exactly solvable models in statistical mechanics. (English) Zbl 0651.57005
The authors present a general method to construct isotopy invariants of classical links from exactly solvable models in statistical mechanics. They show that the Boltzmann weights of such models (which satisfy the Yang-Baxter equation) give rise to representations of the braid groups. The authors specifically consider the Boltzmann weights for the N-state vertex model proposed by K. Sogo, Y. Akutsu, T. Abe in 1983. The authors associate with the new braid group representations the so-called Markov traces and use them to derive (via the Alexander-Markov reduction of links to braids) a series of one-variable polynomial invariants of links. The polynomials corresponding to $$N=2,3,4$$ are treated in some detail. The $$N=2$$ polynomial is the original Jones polynomial. The other polynomials seem to be new. The authors also present a 2-variable extension of the $$N=3$$ polynomial similar to the well known 2-variable extension of the Jones polynomial.
Reviewer’s remark. Essentially the same construction of the isotopy invariants of links from the Yang-Baxter matrices was developed by the reviewer [Invent. Math. 92, 527-553 (1988)].
Reviewer: V.Turaev

##### MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 82B23 Exactly solvable models; Bethe ansatz
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