Explicit R and Racah matrices are given for the symmetric representation of SO(5) to compute Kauffman polynomials via a generalized Reshetikhin-Turaev construction.
Colored knot polynomials. HOMFLY in representation [2,1]
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abstract
This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in various representations a la arXiv:1112.2654, (iii) tabulating and collecting the results at www.knotebook.org. In this paper we discuss only representation R=[2,1] and construct all necessary ingredients that allow one to evaluate knot/links represented by three strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family: this family contains over 80% of knots with up to 10 intersections, but does not include mutants.
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The paper outlines the generalization of cabling-based entangling gates to SU(N) anyons and identifies differences and new problems that arise.
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Racah matrices for the symmetric representation of the SO(5) group
Explicit R and Racah matrices are given for the symmetric representation of SO(5) to compute Kauffman polynomials via a generalized Reshetikhin-Turaev construction.
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Entangling gates for the SU(N) anyons
The paper outlines the generalization of cabling-based entangling gates to SU(N) anyons and identifies differences and new problems that arise.