  In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d
Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way
on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa
partition function becomes a trivial KP tau-function. We study higher genus
corrections to this formula in the form of expansion in powers of z = q-q^{-1}.
Expansion coefficients are expressed through the eigenvalues of the
cut-and-join operators, i.e. symmetric group characters. Moreover, the
z-expansion is naturally exponentiated. Representation through cut-and-join
operators makes contact with Hurwitz theory and its sophisticated integrability
properties. Our formulas describe the shape of genus expansion for the HOMFLY
polynomials, which for their matrix model counterparts is usually controlled by
Virasoro like constraints and AMM/EO topological recursion. The genus expansion
differs from the better studied weak coupling expansion at finite number of
colors N, which is described in terms of the Vassiliev invariants and
Kontsevich integral.
