  In this paper we analyze the quantum homological invariants (the Poincar\'e
polynomials of the $\mathfrak{sl}_N$ link homology). In the case when the
dimensions of homologies of appropriate topological spaces are precisely known,
the procedure of the calculation of the Kovanov-Rozansky type homology, based
on the Euler-Poincar\'e formula can be appreciably simplified. We express the
formal character of the irreducible tensor representation of the classical
groups in terms of the symmetric and spectral functions of hyperbolic geometry.
On the basis of Labastida-Mari\~{n}o-Ooguri-Vafa conjecture, we derive a
representation of the Chern-Simons partition function in the form of an
infinite product in terms of the Ruelle spectral functions (the cases of a
knot, unknot, and links have been considered). We also derive an
infinite-product formula for the orthogonal Chern-Simons partition functions
and analyze the singularities and the symmetry properties of the
infinite-product structures.
