Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid

Abstract

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.