  Character expansion is introduced and explicitly constructed for the
(non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients
are not the knot invariants and can depend on the choice of the braid
realization. However, the method provides the simplest systematic way to
construct HOMFLY polynomials directly in terms of the variable A=q^N: a much
better way than the standard approach making use of the skein relations.
Moreover, representation theory of the simplest quantum group SU_q(2) is
sufficient to get the answers for all braids with m<5 strands. Most important
we reveal a hidden hierarchical structure of expansion coefficients, what
allows one to express all of them through extremely simple elementary
constituents. Generalizations to arbitrary knots and arbitrary representations
is straightforward.
