Tabulating knot polynomials for arborescent knots. (20th January 2017)

• Record Type: Journal Article
• Authors: Mironov, A; Morozov, A; Morozov, A; Ramadevi, P; Singh, Vivek Kumar; Sleptsov, A
• Is Part Of: Journal of physics. Volume 50:Number 8(2017)
• Publication Date: 2017-01-20
• Subjects: Chern–Simons theory -- knot theory -- topological field theory; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aa5574
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: Arborescent knots are those 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 sufficient 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 (http://knotebook.org ). 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, suggested in Mironov and Morozov (2015 Nucl. Phys . B899 395–413), 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-PT polynomials.