Knot polynomials in the first non-symmetric representation

We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure-8 and some other 3-strand knots in representation [21]. This is the first result for non-torus knots beyond (anti)symmetric representations, and its evaluation is far more complicated. We provide a whole variety of different arguments, allowing one to guess the answer for the figure-8 knot, which can be also partly used in more complicated situations. Finally we report the result of exact calculation for figure-8 and some other 3-strand knots based on the previously developed sophisticated technique of multi-strand calculations. We also discuss a formula for the superpolynomial in representation [21] for the figure-8 knot, which heavily relies on the conjectural form of superpolynomial expansion nearby the special polynomial point. Generalizations and details will be presented elsewhere. (C) 2014 The Authors. Published by Elsevier B.V.