Coloured Jones and Alexander polynomials unified through Lagrangian intersections in configuration spaces
Cristina Ana Maria Anghel (University of Oxford)
Monday 9th November, 2020 16:00-17:00 Online
The theory of quantum invariants for knots started with the discovery of the Jones polynomial. After that, Reshetikhin and Turaev developed an algebraic construction which starts with a quantum group and gives an invariant. Using this method, the quantum group Uq(sl(2)) leads to a family of invariants called coloured Jones polynomials. The first term of this sequence is the original Jones polynomial. Dually, the same quantum group at roots of unity leads to a sequence of nonsemisimple quantum invariants, called coloured Alexander polynomials. On the topological side, Lawrence introduced a sequence of homological representations of the braid group, on the homology of coverings of configurations spaces. Then, Bigelow and Lawrence gave a topological model for the original Jones polynomial.
We construct a unified topological model for Uq(sl(2))−quantum invariants. More specifically, we construct certain homology classes in versions of the Lawrence representation, 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 in configuration spaces (which is defined over 3 variables).
The talk will be preceded by a 30 minute tea time. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/91412568415 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).
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