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Title:
DAHA Approach in the Refined Chern-Simons Theory and the BPS Theory of Invariants of Torus Knots
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Abstract:
The Chern-Simons gauge theory (Witten and others) for the Wilson loops associated with the knots on a torus results in the formulas for their Jones and Quantum Group invariants where the key ingredient is the Verlinde S-operator. M.Aganagic and S.Shakirov suggested to replace here S by its “refined” version due to Kirillov and Cherednik in terms of the Macdonald polynomials at roots of unity. They demonstrated that for the torus knots they considered (mainly for the trefoil), such expressions are related to the Poincare polynomials of stable Khovanov-Rozansky triple-graded homology, which presumably coincide with those in the BPS theory. Managing arbitrary torus knots, replacing roots of unity by generic q, the stabilization in terms of the matrix dimension and adding colors were not clarified in this work.
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