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2 results

  • 9 - Macdonald–Koornwinder Polynomials

  • Edited by Tom H. Koornwinder, Universiteit van Amsterdam, Jasper V. Stokman, Universiteit van Amsterdam
  • Book:
    Encyclopedia of Special Functions: The Askey-Bateman Project
    Published online:
    30 September 2020
    Print publication:
    15 October 2020, pp 258-313

  • Summary

    This chapter gives an overview of the theory of nonsymmetric and symmetric Macdonald-Koornwinder polynomials. The setup of the theory is new, allowing for a uniform treatment of all known cases, including a new rank two case. Among the basic properties of the Macdonald-Koornwinder polynomials discussed in the chapter are the (bi)orthogonality relations, the quadratic norm formulas, duality, and evaluation formulas. The chapter also gives an introduction to the associated theory of double affine Hecke algebras.

  • Double affine Hecke algebras and generalized Jones polynomials

  • Part of
  • Yuri Berest, Peter Samuelson
  • Journal:
    Compositio Mathematica / Volume 152 / Issue 7 / July 2016
    Published online by Cambridge University Press:
    01 April 2016, pp. 1333-1384
    Print publication:
    July 2016
    • Article
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  • In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.