On the block structure of the quantum R-matrix in the three-strand braids

Abstract

Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $S{U}_q(N)$ associated with each strand one needs two matrices: ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$. They are related by the Racah matrices ${\mathcal{R}}_2=\mathcal{U}{\mathcal{R}}_1{\mathcal{U}}^{†}$. Since we can always choose the basis so that ${\mathcal{R}}_1$ is diagonal, the problem is reduced to evaluation of ${\mathcal{R}}_2$-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that ${\mathcal{R}}_2$-matrices could be transformed into a block-diagonal ones. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of ${\mathcal{R}}_1$-matrix. The angle of the rotation in the sectors corresponding to accidentally coinciding eigenvalues from the basis defined by the Racah matrix to the basis in which ${\mathcal{R}}_2$ is block-diagonal is $\pm \frac{\pi }{4}$.

arXiv comment21 pages

References in corpus (25)

• Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial
• On Link Invariants and Topological String Amplitudes
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links III : Compact Semi-simple Group
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Three Dimensional Chern-Simons Theory as a Theory of Knots and Links
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Colored HOMFLY polynomials for the pretzel knots and links
• Eigenvalue hypothesis for Racah matrices and HOMFLY polynomials for 3-strand knots in any symmetric and antisymmetric representations
• Knot invariants from rational conformal field theories
• Colored knot polynomials for Pretzel knots and links of arbitrary genus
• Racah coefficients and extended HOMFLY polynomials for all 5-, 6- and 7-strand braids
• Representations of Composite Braids and Invariants for Mutant Knots and Links in Chern-Simons Field Theories
• Chirality of Knots $9_{42}$ and $10_{71}$ and Chern-Simons Theory
• Tabulating knot polynomials for arborescent knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Knot invariants from Virasoro related representation and pretzel knots
• A note on colored HOMFLY polynomials for hyperbolic knots from WZW models
• Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials
• Racah matrices and hidden integrability in evolution of knots
• Quantum Racah matrices and 3-strand braids in irreps R with |R|=4
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Evolution method and HOMFLY polynomials for virtual knots
• Quantum Affine Lie Algebras, Casimir Invariants and Diagonalization of the Braid Generator

Cited by in corpus (7)

• Tangle blocks in the theory of link invariants
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Distinguishing Mutant Knots
• Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• Tug-the-hook symmetry for quantum 6j-symbols
• Multistrand Eigenvalue conjecture and Racah symmetries