Quantum Topology

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Volume 2, Issue 2, 2011, pp. 101–156
DOI: 10.4171/QT/16

Published online: 2011-03-29

Knot polynomial identities and quantum group coincidences

Scott Morrison[1], Emily Peters[2] and Noah Snyder[3]

(1) UC Berkeley, USA
(2) University of New Hampshire, Durham, USA
(3) Columbia University, New York, USA

We construct link invariants using the $\mathcal{D}_{2n}$ subfactor planar algebras, and use these to prove new identities relating certain specializations of colored Jones polynomials to specializations of other quantum knot polynomials. These identities can also be explained by coincidences between small modular categories involving the even parts of the $\mathcal{D}_{2n}$ planar algebras. We discuss the origins of these coincidences, explaining the role of SO level-rank duality, Kirby–Melvin symmetry, and properties of small Dynkin diagrams. One of these coincidences involves $G_2$ and does not appear to be related to level-rank duality.

Keywords: Planar algebras, quantum groups, fusion categories, knot theory, link invariants

Morrison Scott, Peters Emily, Snyder Noah: Knot polynomial identities and quantum group coincidences. Quantum Topol. 2 (2011), 101-156. doi: 10.4171/QT/16