The Sl3 Colored Jones Polynomial of the Trefoil
نویسندگان
چکیده
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of sl3, which allows us to give an explicit formula for the colored Jones polynomial of the trefoil and, more generally, for T (2, n) torus knots. We give two independent proofs of our plethysm formula, one of which uses the work of Carini and Remmel. Our formula for the sl3 colored Jones polynomial of T (2, n) torus knots allows us to verify the Degree Conjecture for those knots, to efficiently determine the sl3 Witten-Reshetikhin-Turaev invariants of the Poincaré sphere, and to guess a Groebner basis for the recursion ideal of the sl3 colored Jones polynomial of the trefoil.
منابع مشابه
THE sl3 JONES POLYNOMIAL OF THE TREFOIL: A CASE STUDY OF q-HOLONOMIC SEQUENCES
The sl3 colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation con...
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