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On the Hecke algebras and the colored HOMFLY polynomial

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arxiv math/0601267 v1 pith:GRLRASSV submitted 2006-01-11 math.QA math.GT

classification math.QAmath.GT
keywords coloredhomflylinkspolynomialformulaquantumalgebrashecke
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abstract

The colored HOMFLY polynomial is the quantum invariant of oriented links in $S^3$ associated with irreducible representations of the quantum group $U_q(\mathrm{sl}_N)$. In this paper, using an approach to calculate quantum invariants of links via cabling-projection rule, we derive a formula for the colored HOMFLY polynomial in terms of the characters of the Hecke algebras and Schur polynomials. The technique leads to a fairly simple formula for the colored HOMFLY polynomial of torus links. This formula allows us to test the Labastida-Mari\~no-Vafa conjecture, which reveals a deep relationship between Chern-Simons gauge theory and string theory, on torus links.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Torus knots in adjoint representation and Vogel's universality

    hep-th 2025-06 conditional novelty 6.0 of 10

    Universal adjoint invariants for T[4,n] torus knots are constructed via Vogel's universality, completing the T[4,n] case after previous T[2,n] and T[3,n] results.

  2. Macdonald deformation of Vogel's universality and link hyperpolynomials

    hep-th 2025-05 conditional novelty 6.0 of 10

    For the adjoint square in ADE Lie algebras, products of Macdonald dimensions with deformed Littlewood-Richardson coefficients are universal, yielding universal formulas for T[2,2n] link hyperpolynomials.

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