Knot polynomial identities and quantum group coincidences

  • Scott Morrison

    UC Berkeley, USA
  • Emily Peters

    University of New Hampshire, Durham, USA
  • Noah Snyder

    Columbia University, New York, USA


We construct link invariants using the D2n\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 D2n\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 G2G_2 and does not appear to be related to level-rank duality.

Cite this article

Scott Morrison, Emily Peters, Noah Snyder, Knot polynomial identities and quantum group coincidences. Quantum Topol. 2 (2011), no. 2, pp. 101–156

DOI 10.4171/QT/16