Monday 12 December 2022
Cristina Palmer-Anghel (Université de Genève / IMAR) — A globalisation of the Jones and Alexander polynomials from configurations on arcs and ovals in the punctured disc
The Jones and Alexander polynomials are two important knot invariants and our aim is to see them from a topological model given by a graded intersection in a configuration space. Bigelow and Lawrence showed a topological model for the Jones polynomial, using arcs and figure eights in the punctured disc. On the other hand, the Alexander polynomial can be obtained from intersections between ovals and arcs.
We present a common topological viewpoint which sees both invariants, based on ovals and arcs in the punctured disc. The model is constructed from a graded intersection between two explicit Lagrangians in a configuration space. It is a polynomial in two variables, recovering the Jones and Alexander polynomials through specialisations of coefficients. Then, we prove that the intersection before specialisation is (up to a quotient) an invariant which globalises these two invariants, given by an explicit interpolation between the Jones polynomial and Alexander polynomial.
We also show how to obtain the quantum generalisation, coloured Jones and coloured Alexander polynomials, from a graded intersection between two Lagrangians in a symmetric power of a surface.
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