$U_q(sl(2))$-quantum invariants from an intersection of two Lagrangians in a symmetric power of a surface

TL;DR

This paper demonstrates that coloured Jones and Alexander polynomials can be derived from a unified geometric framework involving intersections of Lagrangian submanifolds in symmetric powers of a surface, linking knot invariants to symplectic geometry.

Contribution

It introduces a novel geometric interpretation of coloured Jones and Alexander polynomials as graded intersections of explicit Lagrangians in symmetric powers of a surface.

Findings

Coloured Jones and Alexander polynomials are specializations of a graded intersection.

The intersection points are graded using diagonals of the symmetric power.

Original polynomials are special cases involving Heegaard diagrams.

Abstract

In this paper we show that coloured Jones and coloured Alexander polynomials can both be read off from the same picture provided by two Lagrangians in a symmetric power of a surface. More specifically, the $N^{th}$ coloured Jones and $N^{th}$ coloured Alexander polynomials are specialisations of a graded intersection between two explicit Lagrangian submanifolds in a symmetric power of the punctured disc. The graded intersection is parametrised by the intersection points between these Lagrangians, graded in a specific manner using the diagonals of the symmetric power. As a particular case, we see the original Jones and Alexander polynomials as two specialisations of a graded intersection between two Lagrangians in a configuration space, whose geometric supports are Heegaard diagrams.