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Universal coloured Alexander invariant from configurations on ovals in the disc

Cristina Ana-Maria Anghel
Jan 2024
We construct geometrically a {\bf \em universal ADO link invariant} as a limit of {invariants given by graded intersections in configuration spaces}. The question of providing a link invariant that recovers the coloured Alexander invariants for coloured links (which are non-semisimple invariants) was an open problem. A parallel question about semi-simple invariants is the subject of Habiro's famous universal invariants \cite{H3}. First, for a fixed level $\mathcal N$, we construct a link invariant globalising topologically all coloured Alexander link invariants at level less than $\mathcal N$ via the {\bf \em set of intersection points between Lagrangian submanifolds} supported on {\bf \em arcs and ovals} in the disc. Then, based on the naturality of these models when changing the colour, we construct the universal ADO invariant. The purely {\bf \em geometrical origin} of this universal invariant provides a {\bf \em new topological perspective} for the study of the asymptotics of these non-semisimple invariants, for which a purely topological $3$-dimensional description is a deep problem in quantum topology. We finish with a conjecture that our universal invariant has a lift in a module over an extended version of the Habiro ring, which we construct. This paper has a sequel, showing that Witten-Reshetikhin-Turaev and Costantino-Geer-Patureau invariants can both be read off from a fixed set of submanifolds in a configuration space.
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