Uq(sl(2)) quantum invariants for links and 3-manifolds from lagrangian intersections in configuration spaces
2021 07 19
Cristina Ana-Maria Anghel
Knots and representation theory
The theory of quantum invariants started with the Jones polynomial and continued with the Reshetikhin-Turaev algebraic construction. In this context, the quantum group Uq(sl(2)) gives the sequence of coloured Jones polynomials. The quantum group at roots of unity gives the sequence of coloured Alexander polynomials. We construct a uni ed topological model for these two sequences of quantum invariants for links. More specifically, we define certain homology classes given by Lagrangian submanifolds in configuration spaces. Then, we prove that the Nth coloured Jones and Nth coloured Alexander invariants come as different specialisations of a state sum of Lagrangian intersections (de ned over 3 variables) in configuration spaces. As a particular case, we see both Jones and Alexander polynomials from the same intersection pairing in a configuration space. In the second part of the talk we discuss a topological model for the Witten-Reshetikhin-Turaev invariants for 3-manifolds, presenting the N th WRT invariant as a state sum of Lagrangian intersections in a xed configuration space in the punctured disk.
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