Genus expansion of HOMFLY polynomials

In the planar limit of the 't Hooft expansion, the Wilson-loop vacuum average in the three-dimensional Chern–Simons theory (in other words, the HOMFLY polynomial) depends very simply on the representation (Young diagramm), $H_R(A|q)\big|_{q=1}=\bigl(\sigma_1(A)\bigr)^{|R|}$. As a result, the (knot-dependent) Ooguri–Vafa partition function $\sum_RH_R\chi_R\{\bar p_k\}$ becomes a trivial $\tau$-function of the Kadomtsev–Petviashvili hierarchy. We study higher-genus corrections to this formula for $H_R$ in the form of an expansion in powers of $z=q-q^{-1}$. The expansion coefficients are expressed in terms of the eigenvalues of cut-and-join operators, i.e., symmetric group characters. Moreover, the $z$-expansion is naturally written in a product form. The representation in terms of cut-and-join operators relates to the Hurwitz theory and its sophisticated integrability. The obtained relations describe the form of the genus expansion for the HOMFLY polynomials, which for the corresponding matrix model is usually given using Virasoro-like constraints and the topological recursion. The genus expansion differs from the better-studied weak-coupling expansion at a finite number $N$ of colors, which is described in terms of Vassiliev invariants and the Kontsevich integral.