We suggest a relatively simple and totally geometric conjectural description of
uncolored
dahasuperpolynomials of arbitrary algebraic knots (conjecturally
coinciding with the reduced stable Khovanov–Rozansky polynomials) via the flagged
Jacobian factors (new objects) of the corresponding unibranch plane curve
singularities. This generalizes the Cherednik–Danilenko conjecture on the Betti
numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of
superpolynomials of torus knots and that by Gorsky and Mazin for their constant
term. The paper mainly focuses on nontorus algebraic knots. A connection with
the conjecture due to Oblomkov, Rasmussen and Shende is possible, but
our approach is different. A motivic version of our conjecture is related to
–adic orbital
–type
integrals for anisotropic centralizers.
Keywords
Hecke algebra, Jones polynomial, HOMFLYPT polynomial,
Khovanov-Rozansky homology, algebraic knot, Macdonald
polynomial, plane curve singularity, compactified Jacobian,
Puiseux expansion, orbital integral