  Explicit answer is given for the HOMFLY polynomial of the figure eight knot
$4_1$ in arbitrary symmetric representation R=[p]. It generalizes the old
answers for p=1 and 2 and the recently derived results for p=3,4, which are
fully consistent with the Ooguri-Vafa conjecture. The answer can be considered
as a quantization of the \sigma_R = \sigma_{[1]}^{|R|} identity for the
"special" polynomials (they define the leading asymptotics of HOMFLY at q=1),
and arises in a form, convenient for comparison with the representation of the
Jones polynomials as sums of dilogarithm ratios. In particular, we construct a
difference equation ("non-commutative A-polynomial") in the representation
variable p. Simple symmetry transformation provides also a formula for
arbitrary antisymmetric (fundamental) representation R=[1^p], which also passes
some obvious checks. Also straightforward is a deformation from HOMFLY to
superpolynomials. Further generalizations seem possible to arbitrary Young
diagrams R, but these expressions are harder to test because of the lack of
alternative results, even partial.
