A globalisation of Jones and Alexander polynomials constructed from a graded intersection of two Lagrangians in a configuration space

TL;DR

This paper constructs a topological model for the Jones polynomial using graded intersections of Lagrangians in configuration spaces, and introduces a globalisation linking Jones and Alexander polynomials.

Contribution

It provides a topological framework for the Jones polynomial and introduces a globalisation that interpolates between Jones and Alexander polynomials.

Findings

The graded intersection model yields the Jones polynomial as an invariant.

A quadratic relation ensures the invariance of the constructed polynomials.

The globalisation interpolates between Jones and Alexander polynomials within a quotient ring.

Abstract

We consider two Laurent polynomials in two variables associated to a braid, given by {\em graded intersections} between {\em fixed Lagrangians in configuration spaces}. In order to get link invariants, we notice that we have to quotient by a quadratic relation. Then we prove by topological tools that this relation is sufficient and the first graded intersection gives an invariant which is the Jones polynomial. This shows a {\em topological model for the Jones polynomial} and a direct {\em topological proof}\hspace{0.4mm} that it is a well-defined invariant. The other intersection model in the quotient turns out to be an invariant globalising the Jones and Alexander polynomials. This globalisation in the quotient ring is given by a {\em specific interpolation between the Alexander and Jones polynomials}.