  We investigate possibilities of generalizing the TBEM 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.
