Towards matrix model representation of HOMFLY polynomials

We investigate possibilities of generalizing the TBEM (Tierz, Brini–Eynard–Mariño) eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern–Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the $N=2$ case. However, beyond the torus family, both the measure and its lifting to larger $N$ contain non-trivial corrections in $\hbar=\log q$. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz $\tau$-function approach to knot theory, but this remains a subject for future investigation.