Colored knot polynomials. HOMFLY in representation [2,1]

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.

25 pages

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• On rectangular HOMFLY for twist knots
• Differential expansion for link polynomials
• Tangle blocks in the theory of link invariants
• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Eigenvalue hypothesis for multi-strand braids
• Are there p-adic knot invariants?
• Rectangular superpolynomials for the figure-eight knot
• Eigenvalue conjecture and colored Alexander polynomials
• 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 and 3-strand braids in representation [3,3]
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• Gaussian distribution of LMOV numbers
• On the block structure of the quantum R-matrix in the three-strand braids
• On skew tau-functions in higher spin theory
• Wilson Loop Invariants from $W_N$ Conformal Blocks
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• SU(2)/SL(2) knot invariants and KS monodromies
• Vassiliev invariants for pretzel knots
• Vogel's universality and the classification problem for Jacobi identities
• Tug-the-hook symmetry for quantum 6j-symbols
• On ambiguity in knot polynomials for virtual knots
• HOMFLY Polynomials of Pretzel Knots