Quantum Racah matrices up to level 3 and multicolored link invariants

Abstract

This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3⟶Q$ with all possible $Q$, for $|R|\le 3$. 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 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.

arXiv comment29 pages, 3 figures

References in corpus (48)

• Knot Invariants and Topological Strings
• Knots, links and branes at large N
• Polynomial invariants for torus knots and topological strings
• 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
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Character expansion for HOMFLY polynomials. III. All 3-Strand braids in the first symmetric representation
• Evolution method and "differential hierarchy" of colored knot polynomials
• Colored HOMFLY polynomials for the pretzel knots and links
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• Character expansion for HOMFLY polynomials. I. Integrability and difference equations
• Knots-quivers correspondence
• 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
• BPS states, knots and quivers
• Chern-Simons Invariants of Torus Links
• Multiplicity-free quantum 6j-symbols for U_q(sl_N)
• Super-A-polynomials for Twist Knots
• Towards effective topological field theory for knots
• 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]
• String theory and the Kauffman polynomial
• Towards topological quantum computer
• Colored HOMFLY polynomials that distinguish mutant knots
• Colored Kauffman Homology and Super-A-polynomials
• Reformulated invariants for non-torus knots and links
• Racah matrices and hidden integrability in evolution of knots
• Knots, BPS states, and algebraic curves
• Link polynomial calculus and the AENV conjecture
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Cabling procedure for the colored HOMFLY polynomials
• BPS counting for knots and combinatorics on words
• Differential expansion for link polynomials
• Eigenvalue hypothesis for multi-strand braids
• On R-matrix approaches to knot invariants
• Checks of integrality properties in topological strings
• Integrality structures in topological strings I: framed unknot
• Refined large N duality for knots
• Quantum Racah matrices and 3-strand braids in representation [3,3]
• Colored HOMFLY polynomials from Chern-Simons theory
• Gaussian distribution of LMOV numbers
• Composite Representation Invariants and Unoriented Topological String Amplitudes
• Composite Invariants and Unoriented Topological String Amplitudes
• Proof of the Labastida-Marino-Ooguri-Vafa Conjecture

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• Tangle blocks in the theory of link invariants
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Distinguishing Mutant Knots
• New symmetries for the $U_q(s{l}_N)$ 6-j symbols from the Eigenvalue conjecture
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• On measuring the topological charge of anyons
• Tug-the-hook symmetry for quantum 6j-symbols
• Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism