Chern-Simons perturbative series revisited

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

“…The knot dependent functions V K n,m are called Vassiliev invariants or finite type invariants [27,28]. The group dependent functions G R n,m are called group factors, and dim G n is a number of linearly independent group factors G R n,m at a certain level n. While the structure of Vassiliev invariants remains mysterious, in previous work [1] we have closely approached a complete description of the HOMFLY group structure and provided an algorithm to compute the group factors that are valid for any representation R and an arbitrary value of N . This paper is devoted to applications of the new knowledge on group factors to the research of new properties of quantum knot invariants.…”

“…First, many knot polynomials in different (large enough) representations are known. Second, an explicit form of the corresponding group factors has been obtained [1]. An explicit form of the HOMFLY group factors was available only for sl N fundamental representation and any sl 2 representation [28], so that Vassiliev invariants were obtained only up to the 6-th order [29].…”

“…An explicit form of the HOMFLY group factors was available only for sl N fundamental representation and any sl 2 representation [28], so that Vassiliev invariants were obtained only up to the 6-th order [29]. Our result [1] allows one to compute Vassiliev invariants of higher orders (see Section 4), which is crucial in some cases; for example, mutant knots [30][31][32] have the same Vassiliev invariants up to the 10-th order.…”

“…The new algorithm reduces a computation of a group factor to a simple combinatorial problem. It has two advantages compared with the previous method [1]. First, the new algorithm provides a unified approach to all group factor.…”

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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.

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