  We continue the program of systematic study of extended HOMFLY polynomials.
Extended polynomials depend on infinitely many time variables, are close
relatives of integrable tau-functions, and depend on the choice of the braid
representation of the knot. They possess natural character decompositions, with
coefficients which can be defined by exhaustively general formula for any
particular number m of strands in the braid and any particular representation R
of the Lie algebra GL(\infty). Being restricted to "the topological locus" in
the space of time variables, the extended HOMFLY polynomials reproduce the
ordinary knot invariants. We derive such a general formula, for m=3, when the
braid is parameterized by a sequence of integers (a_1,b_1,a_2,b_2,...), and for
the first non-fundamental representation R=[2]. Instead of calculating the
mixing matrices directly, we deduce them from comparison with the known answers
for torus and composite knots. A simple reflection symmetry converts the answer
for the symmetric representation [2] into that for the antisymmetric one [1,1].
The result applies, in particular, to the figure eight knot 4_1, and was
further extended to superpolynomials in arbitrary symmetric and antisymmetric
representations in arXiv:1203.5978.
