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

  • Igor B. Frenkel
  • Vladimir G. Turaev
Part of the Progress in Mathematics book series (PM, volume 131/132)


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
$${R_{12}}{R_{23}}{R_{12}} = {R_{23}}{R_{12}}{R_{23}}\left( {0,a} \right)$$
where R is an endomorphism of VV 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 ℝ3 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.


Tral Mirror Symmetry 


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Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Igor B. Frenkel
    • 1
  • Vladimir G. Turaev
    • 2
  1. 1.Department of MathematicsYale UniversityNew HavenUSA
  2. 2.Department of MathematicsLouis Pasteur University - CNRSStrasbourgFrance

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