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

Abstract

We have recently proposed arXiv:2105.11565 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 $R$.

24 pages

References in corpus (9)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Difference equation of the colored Jones polynomial for torus knot
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• Differential expansion for link polynomials
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• Chern-Simons perturbative series revisited
• Difference of mutant knot invariants and their differential expansion
• A note on the gl(m|n) link invariants and the HOMFLY-PT polynomial

Cited by in corpus (5)

• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Evolution properties of the knot's defect
• Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism
• Machine learning of the well known things
• Algebraic structures of Vassiliev invariants for knot families