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A note on colored HOMFLY polynomials for hyperbolic knots from WZW models. (English) Zbl 1328.81193
Summary: Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models, we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY invariants in terms of crossing matrices of the underlying Wess-Zumino-Witten model. Our analysis extends previous works by incorporating non-trivial multiplicities for the primaries appearing in the crossing matrices, so as to describe colorings of HOMFLY invariants beyond the totally symmetric or anti-symmetric representations of \(\mathrm{SU}(N)\). The crossing matrices directly relate to 6j-symbols of the quantum group \(\mathcal{U}_q\mathfrak{su}(N)\). We present powerful methods to calculate such quantum 6j-symbols for general N. This allows us to determine previously unknown colored HOMFLY polynomials for two-bridge hyperbolic knots. We give explicitly the HOMFLY polynomials colored by the representation \(\{2, 1\}\) for two-bridge hyperbolic knots with up to eight crossings. Yet, the scope of application of our techniques goes beyond knot theory; e.g., our findings can be used to study correlators in Wess-Zumino-Witten conformal field theories or – in the limit to classical groups – to determine color factors for Yang Mills amplitudes.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
58J28 Eta-invariants, Chern-Simons invariants
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
81T45 Topological field theories in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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