Cabling procedure for the colored HOMFLY polynomials

Abstract

In the present paper we discuss the cabling procedure for the colored HOMFLY polynomial. We describe how it can be used and how one can find all the quantities such as projectors and $\mathcal{R}$-matrices, which are needed in this procedure. The constructed matrix forms of the projectors and the fundamental $\mathcal{R}$-matrices allow one in principle (neglecting the computational difficulties) to find the HOMFLY polynomial in any representation for any knot. We also discuss the group theory explanation of the cabling procedure. This leads to the explanations of the form of the fundamental $\mathcal{R}$-matrices and illuminates several conjectures proposed in previous papers.

52 pages + Tables of Colored Knot Polynomials

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• Colored HOMFLY polynomials for the pretzel knots and links
• On colored HOMFLY polynomials for twist knots
• Tabulating knot polynomials for arborescent knots
• Colored knot polynomials. HOMFLY in representation [2,1]
• Link polynomial calculus and the AENV conjecture
• On rectangular HOMFLY for twist knots
• Differential expansion for link polynomials
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Chern-Simons perturbative series revisited
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• SU(2)/SL(2) knot invariants and KS monodromies
• On measuring the topological charge of anyons
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• Torus Knots in Adjoint Representation
• Colored HOMFLY and Generalized Mandelbrot set