# On the interaction between the Reshetikhin-Turaev invariant and Springer theory

## Seminar/Forum

107
Peter Hall

1) The Reshetikhin-Turaev invariant provides an invariant of tangles for each choice of a simple Lie algebra together with a representation of it for each component of the tangle. For $$sl_ 2$$ and its standard representation the invariant coincides with the well-known Jones polynomial when restricted to knots and links.
The tool allowing us to pass back and forth between 1) and 2) is given by Schur-Weyl duality. By going from 1) to 2) we obtain an explicit understanding of the topology of certain Springer fibers using combinatorics extracted from the Reshetikhin-Turaev invariant for $$sl_ 2$$. By going from 2) to 1) we can use Springer fibers to categorify the Reshetikhin-Turaev invariant for $$sl_ 2$$ and recover Khovanov homology in a geometric disguise.