  The Schur functions play a crucial role in the modern description of HOMFLY
polynomials for knots and of topological vertices in DIM-based network
theories, which could merge into a unified theory still to be developed. The
Macdonald functions do the same for hyperpolynomials and refined vertices, but
merging appears to be more problematic. For a detailed study of this problem,
more knowledge is needed about the Macdonald polynomials than is usually
available. As a preparation for the discussion of the knot/vertices relation,
we summarize the relevant facts and open problems about the Macdonald and, more
generally, Kerov functions. Like Macdonald polynomials, they are triangular
combinations of Schur functions, but orthogonal in a more general scalar
product. We explain that the parameters of the measure can be considered as a
set of new time variables, and the Kerov functions are actually expressed
through the Schur functions of these variables as well. Despite they provide an
infinite-parametric extension of the Schur and Macdonald polynomials, the Kerov
functions, and even the skew Kerov functions continue to satisfy the most
important relations, like Cauchy summation formula, the transposition identity
for reflection of the Young diagram and expression of the skew functions
through the generalized Littlewood-Richardson structure constants. Since {\it
these} are the properties important in most applications, one can expect that
the Kerov extension exists for most of them, from the superintegrable matrix
and tensor models to knot theory.
