Torus knots and the rational DAHA

Abstract

We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebraic expression in this picture, yielding a prescription for the doubly graded Khovanov-Rozansky homologies. We match our conjecture to previous conjectures of the first author relating knot homology to q, t-Catalan numbers, and of the last three authors relating knot homology to Hilbert schemes on singular curves.

54 pages

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Cited by in corpus (21)

• The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link
• Toric braids and $(m,n)$-parking functions
• Sequencing BPS Spectra
• DAHA and iterated torus knots
• On the computation of torus link homology
• Categorified Young symmetrizers and stable homology of torus links
• An Exceptional Collection For Khovanov Homology
• On Factorization of Generalized Macdonald Polynomials
• Condensates and instanton - torus knot duality. Hidden Physics at UV scale
• The Condensate from Torus Knots
• On stable sl3-homology of torus knots
• DAHA and plane curve singularities
• Modules over plane curve singularities in any ranks and DAHA
• A support theorem for Hilbert schemes of planar curves, II
• Iterated torus knots and double affine Hecke algebras
• Representations with minimal support for quantized Gieseker varieties
• Nef cones of nested Hilbert schemes of points on surfaces
• From representations of the rational Cherednik algebra to parabolic Hilbert schemes via the Dunkl-Opdam subalgebra
• Flops and Hilbert schemes of space curve singularities
• The Hilb-vs-Quot Conjecture
• Superpolynomials of algebraic links