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

Understand

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.