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

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

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

Keywords

Nonnegative Integer Spectral Parameter Hypergeometric Function Jones Polynomial Reidemeister Move 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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