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Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid. (English) Zbl 1309.81114
Summary: [For part I see the authors, arxiv:1112.5754, 12 p. (2011)] Character expansion is introduced and explicitly constructed for the (non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients are not the knot invariants and can depend on the choice of the braid realization. However, the method provides the simplest systematic way to construct HOMFLY polynomials directly in terms of the variable \(A = q^{N}\): a much better way than the standard approach making use of the skein relations. Moreover, representation theory of the simplest quantum group \(\operatorname{SU}_{q}(2)\) is sufficient to get the answers for all braids with \(m < 5\) strands. Most important we reveal a hidden hierarchical structure of expansion coefficients, what allows one to express all of them through extremely simple elementary constituents. Generalizations to arbitrary knots and arbitrary representations is straightforward.

MSC:
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81T75 Noncommutative geometry methods in quantum field theory
58J28 Eta-invariants, Chern-Simons invariants
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
Software:
Knot Atlas
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