Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

Abstract

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU (N ) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T [2, 2k + 1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

Evolution properties of the knot’s defect

“…Despite the simplicity of definitions, this expansion is rather difficult to get and study even in particular examples, and we are still at the stage when new examples provide new insights. For recent developments in another direction -perturbative expansion of HOMFLY-PT polynomials, see [37,38].…”