Tabulating knot polynomials for arborescent knots

Abstract

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.

19 pages

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• Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations
• Multiplicity-free $U_q(s{l}_N)$ 6-j symbols: relations, asymptotics, symmetries
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• Distinguishing Mutant Knots
• Perspectives of differential expansion
• Extension of KNTZ trick to non-rectangular representations
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• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• Quantum Racah matrices up to level 3 and multicolored link invariants
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• Gaussian distribution of LMOV numbers
• On the block structure of the quantum R-matrix in the three-strand braids
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• HOMFLY for twist knots and exclusive Racah matrices in representation [333]
• KNTZ trick from arborescent calculus and the structure of differential expansion
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• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
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• Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed
• Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure
• On measuring the topological charge of anyons
• Algebra of quantum $\mathcal{C}$-polynomials
• Evolution properties of the knot's defect
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• Vogel's universality and the classification problem for Jacobi identities
• Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism
• Entangling gates from cabling of knots