  Knot matrix models are defined so that the averages of characters are equal
to knot polynomials. From this definition one can extract single trace averages
and generation functions for them in the group rank - which generalize the
celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the
outcome of this program for HOMFLY-PT polynomials of various knots. In
particular, we claim that the Harer-Zagier formulas for torus knots factorize
nicely, but this does not happen for other knots. This fact is mysteriously
parallel to existence of explicit beta = 1 eigenvalue model construction for
torus knots only, and can be responsible for problems with construction of a
similar model for other knots.
