Harer-Zagier formulas for knot matrix models

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.