# Trigonometric Solutions of the Yang-Baxter Equation, Nets, and Hypergeometric Functions

Chapter

## Abstract

It is now well known that the topology of knots in Euclidean 3-space is closely related to the study of the quantum Yang-Baxter equation
where

$${R_{12}}{R_{23}}{R_{12}} = {R_{23}}{R_{12}}{R_{23}}\left( {0,a} \right)$$

(0.a)

*R*is an endomorphism of*V*⊗*V*for a vector space*V*. The solutions of this equation, i.e.,*R*-matrices, give rise to topological invariants of knots. To compute the invariant determined by*R*, we present an oriented knot in ℝ^{3}by a diagram in the plane and assign*R*(resp. R^{-1}) to every positive (resp. negative) crossing of the diagram. Contracting these matrices along the 1-strata of the diagram, we obtain a number. The Yang-Baxter equation guarantees that this number is preserved under the third Reidemeister move on knot diagrams. Under favorable conditions, this number is preserved under other Reidemeister moves and yields a topological invariant of knots. In this way we can obtain the Jones polynomial of knots and its various generalizations. This theory has a number of important ramifications, including a computation of the invariant in terms of 6j-symbols. Moreover, this theory leads to invariants of 3-manifolds, 2-dimensional modular functors, and 3-dimensional topological quantum field theories; see [Tu] and references therein.### Keywords

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### References

- [Ba]R. J. Baxter,
*Exactly Solved Models in Statistical Mechanics*, Academic Press, 1982Google Scholar - [GR]G. Gasper, M. Rahman,
*Basic Hypergeometric Series*, Cambridge Univ. Press, 1990Google Scholar - [DJMO]E. Date, M. Jimbo, T. Miwa, M. Okado, Fusion of the eight-vertex
*SOS*model,*Lett. Math. Phys*.**12**(1986), 209–215. Erratum and Addendum:*Lett. Math. Phys.***14**(1987), 97MathSciNetCrossRefGoogle Scholar - [FT]I. B. Frenkel, V. G. Turaev, Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions, to appearGoogle Scholar
- [Ji]M.Jimbo, Solvable lattice models and quantum groups,
*Proceedings of the ICM*,Kyoto, Japan, 1990, Springer-Verlag, 1343–1352Google Scholar - [Jo]V.F.R. Jones, Index for subfactors,
*Invent. Math.***72**(1983), 1–25MathSciNetMATHCrossRefGoogle Scholar - [KL]L. Kauffman, S.L. Lins,
*Temperley-Lieb recoupling theory and invari-**ants of 3-manifolds*, Princeton Univ. Press, Princeton, N.J. 1994Google Scholar - [KR]A.N. Kirillov, N.Y. Reshetikhin, Representations of the algebra
*U**q**(sl**2**)*, q-orthogonal polynomials and invariants of links. In:*Infinite dimensional Lie algebras and groups*, (ed. V.G. Kac), 285–339. Adv. Ser. in Math. Phys. 7, World Scientific, Singapore 1988Google Scholar - [KRS]P.P. Kulish, N.Y. Reshetikhin, E.K. Sklyanin, Yang-Baxter equation and representation theory,
*I. Lett. Math. Phys.***5**(1981), 393–403MathSciNetMATHCrossRefGoogle Scholar - [MV]G. Masbaum, P. Vogel, 3-valent graphs and the Kauffman bracket,
*Pac. J. Math.***164**(1994), 361–381MathSciNetMATHGoogle Scholar - [Tu]V. G. Turaev,
*Quantum Invariants of Knots and 3-Manifolds*, de Gruyter Studies in Math. 18, 1994Google Scholar - [We]H. Wenzl, On sequences of projections,
*C. R. Math. Rep. Acad. Sci*. Canada 9, n. 1 (1987), 5–9MathSciNetMATHGoogle Scholar

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