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Quantum Racah matrices up to level 3 and multicolored link invariants. (English) Zbl 1397.57023
Summary: This paper is a next step in the project of systematic description of colored knot and link invariants started in [A. Mironov and A. Morozov, Nucl. Phys., B 899, 395–413 (2015; Zbl 1331.81264) and A. Mironov et al., J. Phys. A, Math. Theor. 50, No. 8, Article ID 085201, 22 p. (2017; Zbl 1360.81271)]. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $$R_1 \otimes R_2 \otimes R_3 \longrightarrow Q$$ with all possible $$Q$$, for $$| R | \leq 3$$. The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of a trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.

MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 81T45 Topological field theories in quantum mechanics 81P68 Quantum computation
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