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DAHA-Jones polynomials of torus knots

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Abstract

DAHA-Jones polynomials of torus knots T(r, s) are studied systematically for reduced root systems and in the case of \(C^\vee C_1\). We prove the polynomiality and evaluation conjectures from the author’s previous paper on torus knots and extend the theory by the color exchange and further symmetries. The DAHA-Jones polynomials for \(C^\vee C_1\) depend on five parameters. Their surprising connection to the DAHA-superpolynomials (type A) for the knots \(T(2p+1,2)\) is obtained, a remarkable combination of the color exchange conditions and the author’s duality conjecture (justified by Gorsky and Negut). The uncolored DAHA-superpolynomials of torus knots are expected to coincide with the Khovanov–Rozansky stable polynomials and the superpolynomials defined via rational DAHA and/or in terms of certain Hilbert schemes. We end the paper with certain arithmetic counterparts of DAHA-Jones polynomials for the absolute Galois group in the case of \(C^\vee C_1\), developing the author’s previous results for \(A_1\).

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Acknowledgments

The paper was mainly written at RIMS; the author thanks Hiraku Nakajima and RIMS for the invitation and hospitality and the Simons Foundation (which made this visit possible).

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Authors and Affiliations

  1. Department of Mathematics, UNC Chapel Hill, Chapel Hill, NC, 27599, USA

    Ivan Cherednik

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  1. Ivan Cherednik
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Correspondence to Ivan Cherednik.

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Partially supported by NSF Grant DMS—1363138 and the Simons Foundation.

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Cherednik, I. DAHA-Jones polynomials of torus knots. Sel. Math. New Ser. 22, 1013–1053 (2016). https://doi.org/10.1007/s00029-015-0210-1

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