Geometric universal Jones invariant from configurations on ovals in the disc (2401.17245v2)

Abstract: We construct geometrically a universal Jones invariant as a limit of invariants given by graded intersections in configuration spaces. For any fixed level $\mathcal N$, we construct a new knot invariant, called "$\mathcal N^{th}$ Unified Jones invariant" globalising topologically all coloured Jones polynomials at levels less than $\mathcal N$. It is defined via the set of intersection points between Lagrangian submanifolds supported on arcs and ovals in the disc. Habiro's famous invariant for knots is a universal invariant recovering each coloured Jones polynomial. Willetts modified Habiro's ring and defined a universal invariant recovering coloured Jones polynomials and the ADO invariant divided by the Alexander polynomials. The universal Jones invariant that we construct belongs to a different ring that comes with a map to Habiro-Willetts's ring. We prove that our invariant recovers Habiro-Willetts's invariant. The difference is that our invariant is constructed as a limit of new knot invariants, the $\mathcal N^{th}$ unified Jones invariants. These invariants in turn provide a geometrical understanding of sets of all coloured Jones polynomials of bounded colour, collecting more information as we increase the colour. This article provides part of the set-up for a sequel, where we construct geometrically two universal link invariants. These invariants, universal Jones link invariant and universal ADO link invariant, globalise all coloured Jones polynomials and all ADO link invariants respectively. They are constructed as limits of link invariants that see all coloured Jones link invariants and coloured Alexander link invariants with bounded (multi) colours respectively. This answers the open problem of defining universal non-semisimple link invariants.