  Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY
polynomials is the Hurwitz partition function, i.e. that the dependence of the
HOMFLY polynomials on representation R is naturally captured by symmetric group
characters (cut-and-join eigenvalues). The genus expansion and expansion
through Vassiliev invariants explicitly demonstrate this phenomenon. In the
present letter we claim that the superpolynomials are not functions of such a
type: symmetric group characters do not provide an adequate linear basis for
their expansions. Deformation to superpolynomials is, however, straightforward
in the multiplicative basis: the Casimir operators are \beta-deformed to
Hamiltonians of the Calogero-Moser-Sutherland system. Applying this trick to
the genus and Vassiliev expansions, we observe that the deformation is fully
straightforward only for the thin knots. Beyond the family of thin knots
additional algebraically independent terms appear in the Vassiliev and genus
expansions. This can suggest that the superpolynomials do in fact contain more
information about knots than the colored HOMFLY and Kauffman polynomials.
However, even for the thin knots the beta-deformation is non-innocent: already
in the simplest examples it seems inconsistent with the postivity of colored
superpolynomials in non-(anti)symmetric representations, which also happens in
I.Cherednik's (DAHA-based) approach to the torus knots.
