HOMFLY polynomials in representation [3,1] for 3-strand braids

Abstract

This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3,1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n,-1|1,-1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.

21 pages

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Cited by in corpus (19)

• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• On rectangular HOMFLY for twist knots
• Differential expansion for link polynomials
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Eigenvalue hypothesis for multi-strand braids
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities
• Rectangular superpolynomials for the figure-eight knot
• Perspectives of differential expansion
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• Orthogonal Polynomials in Mathematical Physics
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Quantum Racah matrices and 3-strand braids in representation [3,3]
• Gaussian distribution of LMOV numbers
• On the block structure of the quantum R-matrix in the three-strand braids
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Tug-the-hook symmetry for quantum 6j-symbols