DAHA and iterated torus knots

Abstract

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov-Shende-Rasmussen Conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a=0, q=1 are conjectured to provide the Betti numbers of the Jacobian factors of the corresponding singularities.

arXiv commentv2: Editing, more material on the Betti numbers of Jacobian factors and non-algebraic knots was added, the relation to Jones polynomials was justified

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• Macdonald's Evaluation Conjectures, Difference Fourier Transform, and Applications
• Homological algebra of knots and BPS states
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• DAHA-Jones polynomials of torus knots
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Cited by in corpus (25)

• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Colored HOMFLY polynomials for the pretzel knots and links
• On universal knot polynomials
• Toric braids and $(m,n)$-parking functions
• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra
• Homology of torus knots
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• Rectangular superpolynomials for the figure-eight knot
• Knot polynomials for twist satellites
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• A non-torus link from topological vertex
• On matrix-model approach to simplified Khovanov-Rozansky calculus
• Genus Two Generalization of $A_1$ spherical DAHA
• DAHA and plane curve singularities
• Iterated torus knots and double affine Hecke algebras
• Modules over plane curve singularities in any ranks and DAHA
• On the genus two skein algebra
• Chaos and integrability in SL(2,R)-geometry
• Hopf superpolynomial from topological vertices
• Matrix model and dimensions at hypercube vertices
• Cyclotomic Expansion of Generalized Jones Polynomials
• Superpolynomials of algebraic links