Universal geometrical link invariants

TL;DR

This paper constructs universal link invariants that unify various colored Jones and ADO polynomials, providing new semi-simple and non semi-simple invariants through geometric and topological methods.

Contribution

It introduces the first universal non semi-simple link invariant unifying all ADO invariants, and a new semi-simple universal link invariant for all links.

Findings

Constructed universal ADO and Jones invariants as limits of configuration space intersections.

Unified all colored Jones polynomials and ADO polynomials within a geometric framework.

Provided explicit geometric constructions for these universal invariants.

Abstract

We construct geometrically two universal link invariants: universal ADO invariant and universal Jones invariant, as limits of invariants given by graded intersections in configuration spaces. More specifically, for a fixed level $\mathscr N$, we define new link invariants: ``$\mathscr N^{th}$ Unified Jones invariant'' and ``$\mathscr N^{th}$ Unified Alexander invariant''. They globalise topologically all coloured Jones polynomials for links with multi-colours bounded by $\mathscr N$ and all ADO polynomials with bounded colours. These invariants both come from the same weighted Lagrangian intersection supported on configurations on arcs and ovals in the disc. The question of providing a universal non semi-simple link invariant, recovering all the ADO polynomials was an open problem. A parallel question about semi-simple invariants for the case of knots is the subject of Habiro's famous universal knot invariant \cite{H3}. Habiro's universal construction is well defined for knots and can be extended just for certain classes of links. Our universal Jones link invariant is defined for any link and recovers all coloured Jones polynomials, providing a new semi-simple universal link invariant. The first non semi-simple universal link invariant that we construct unifies all ADO invariants for links, answering the open problem about the globalisation of these invariants.