On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions

A. Mironov, A. Morozov, A. Sleptsov
2013 European Physical Journal C: Particles and Fields  
In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential tau-function. In fact, in the double scaling Kashaev limit (the one associated with the volume conjecture) dominant in the genus expansion are terms associated with
more » ... he symmetric representations and with the integrability preserving Casimir operators, though we stop one step from converting this fact into a clear statement about the OVPF behavior in the vicinity of q=1. Instead, we explain that the genus expansion provides a hierarchical decomposition of the Hurwitz tau-function, similar to the Takasaki-Takebe expansion of the KP tau-functions. This analogy can be helpful to develop a substitute for the universal Grassmannian description in the Hurwitz tau-functions.
doi:10.1140/epjc/s10052-013-2492-9 fatcat:3uiezyoqore4dekrlucerj3smq