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Article

Factorization of colored knot polynomials at roots of unity

Kononov Y., Morozov A.

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) 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=1, HOMFLY polynomials in symmetric representations [r] satisfy recursion identity: Hr+m=Hr{dot operator}Hm for any A=qN, which is a generalization of the property Hr=H1r for special polynomials at m=1. We conjecture a further generalization to arbitrary representation R, which, however, is checked only for torus knots. Next, Kashaev polynomial, which arises from HR at q2=e2πi/|R|, turns equal to the special polynomial with A substituted by A|R|, provided R is a single-hook representations (including arbitrary symmetric) - what provides a q-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. © 2015 Published by Elsevier B.V.