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HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

  • H. Itoyama
  • A. Mironov
  • A. Morozov
  • And. Morozov
Article

Abstract

Explicit answer is given for the HOMFLY polynomial of the figure eight knot 41 in arbitrary symmetric representation R = [p]. It generalizes the old answers for p = 1 and 2 and the recently derived results for p = 3, 4, which are fully consistent with the Ooguri-Vafa conjecture. The answer can be considered as a quantization of the Open image in new window identity for the “special” polynomials (they define the leading asymptotics of HOMFLY at q = 1), and arises in a form, convenient for comparison with the representation of the Jones polynomials as sums of dilogarithm ratios. In particular, we construct a difference equation (“non-commutative \( \mathcal{A} \)-polynomial”) in the representation variable p. Simple symmetry transformation provides also a formula for arbitrary antisymmetric (fundamental) representation R = [1 p ], which also passes some obvious checks. Also straightforward is a deformation from HOMFLY to superpolynomials. Further generalizations seem possible to arbitrary Young diagrams R, but these expressions are harder to test because of the lack of alternative results, even partial.

Keywords

Chern-Simons Theories Topological Strings Quantum Groups 

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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • H. Itoyama
    • 1
  • A. Mironov
    • 2
    • 3
  • A. Morozov
    • 3
  • And. Morozov
    • 3
    • 4
  1. 1.Department of Mathematics and PhysicsOsaka City University and Osaka City University Advanced Mathematical Institute (OCAMI)OsakaJapan
  2. 2.Theory DepartmentLebedev Physics InstituteMoscowRussia
  3. 3.Institute for Theoretical and Experimental PhysicsMoscowRussia
  4. 4.Physics FacultyMoscow State UniversityMoscowRussia

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