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A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models

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Abstract

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 SU(N). The crossing matrices directly relate to 6j-symbols of the quantum group \({\mathcal{U}_{q}su(N)}\). 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.

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  1. Bethe Center for Theoretical Physics, Physikalisches Institut, Universität Bonn, 53115, Bonn, Germany

    Jie Gu & Hans Jockers

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  1. Jie Gu
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  2. Hans Jockers
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Communicated by H. Ooguri

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Gu, J., Jockers, H. A Note on Colored HOMFLY Polynomials for Hyperbolic Knots from WZW Models. Commun. Math. Phys. 338, 393–456 (2015). https://doi.org/10.1007/s00220-015-2322-z

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