Geometric universal Jones invariant from configurations on ovals in the disc

Abstract

We construct geometrically a universal Jones invariant as a limit of invariants given by graded intersections in configuration spaces. For any fixed level $\mathscr N$, we define a new knot invariant, called ``$\mathscr N^{th}$ Unified Jones invariant'' globalising topologically all coloured Jones polynomials at levels less than $\mathscr N$. It is defined via the intersection points between {Lagrangian submanifolds} supported on arcs and ovals in the disc. The geometry of these Lagrangians is novel: previous topological models involved immersed submanifolds rather than embedded ones. We do this by defining a new local system that refines the Lawrence representation, and depends of the distribution of multiplicities of points in the configuration space on the ovals. On the algebraic side, Habiro's famous invariant for knots \cite{H3} is a universal invariant globalising the family of coloured Jones polynomials. He conjectured that this universal invariant recovers also the ADO invariant divided by the Alexander polynomials, which was proved by Willetts \cite{W} in a version of Habiro's ring (\cite{H2}). The universal Jones invariant that we construct belongs to a different ring that comes with a map to Habiro's ring \cite{H2}. We prove that our invariant recovers this version of Habiro's invariant. The difference is that our invariant is given as a limit of new knot invariants, the $\mathscr N^{th}$ 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.