Factorization of colored knot polynomials at roots of unity

HOMFLY polynomials are the Wilson-loop averages in Chern–Simons theory and depend on four variables: the closed line (knot) in 3d space–time, representation R   of the gauge group SU(N)SU(N) and exponentiated coupling constant q. From analysis of a big variety of different knots we conclude that at q, which is a 2m  -th root of unity, q2m=1q2m=1, HOMFLY polynomials in symmetric representations [r][r] satisfy recursion identity: Hr+m=Hr⋅HmHr+m=Hr⋅Hm for any A=qNA=qN, which is a generalization of the property Hr=H1r for special polynomials at m=1m=1. We conjecture a further generalization to arbitrary representation R  , which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HRHR at q2=e2πi/|R|q2=e2πi/|R|, turns equal to the special polynomial with A   substituted by A|R|A|R|, provided R   is a single-hook representations (including arbitrary symmetric) – what provides a q−Aq−A dual to the similar property of Alexander polynomial. All this implies non-trivial relations for the coefficients of the differential expansions, which are believed to provide reasonable coordinates in the space of knots – existence of such universal relations means that these variables are still not unconstrained.