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Colored knot polynomials: HOMFLY in representation \([2, 1]\). (English) Zbl 1333.81202
The authors propose knot polynomials to be correlators of Wilson line in 3D Chern-Simons theory. For example, the most important HOMFLY polynomials are
\[ H^{\mathcal{L}\subset\mathcal{M}}_R(q,A)=\langle\mathrm{Tr}_RP\exp\oint_\mathcal{L}\mathcal{A}\rangle^{\mathrm{CS}}. \] Here, \(\mathcal{L}\) is a knot (or link) in \(\mathcal{M}\), a three dimensional space, \(q=\exp(\frac{2\pi i}{k+N})\) is the CS coupling constant and \(R\) is the representation (Young diagram) of \(G=\mathrm{SU}(N)\). This paper aims to classify HOMFLY polynomials in representation \(R=[2,1]\). The classification of knots follows from the authors previous proposal that represent the knots and links as the two-bridges “finger” and “propagators” [the first author et al., J. High Energy Phys. 2015, No. 7, Article ID 109, 70 p. (2015; doi:10.1007/JHEP07(2015)109)].
After computing Racah matrices, which transform an orthonormal basis of \((R\otimes R)\otimes R\), \(R=[2,1]\) to an orthonormal basis of \(R\otimes(R\otimes R)\) in a vector space \(Q\), adopting a brute force method of [the first author et al., J. High Energy Phys. 2012, No. 3, Article ID 034, 34 p. (2012; Zbl 1309.81114)], clasification and explicit computations of HOMFLY polynomials are given.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
58J28 Eta-invariants, Chern-Simons invariants
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