A Geometric Approach to the Links-Quivers Correspondence II: Rational Links

TL;DR

This paper provides a geometric description of the quiver form for the HOMFLY-PT polynomials of rational links, connecting link invariants with configuration spaces of the 3-punctured plane.

Contribution

It offers a direct geometric interpretation of the linear and quadratic forms in the quiver representation for rational links, extending previous algebraic results.

Findings

Geometric description of the linear and quadratic forms in quiver form

Connection between HOMFLY-PT polynomials and configuration spaces

Extension of previous proofs to a geometric framework

Abstract

The Links-Quivers Correspondence predicts that the generating function for the symmetric (or antisymmetric) colored HOMFLY-PT polynomials for links can be put in a "quiver form," so that the generating function is expressed in terms of a quadratic form and two linear forms. This was originally proved for rational links by Stosic and Wedrich, but here we give a direct geometric description of the linear and quadratic forms in terms of the first and second configuration spaces of the 3-punctured plane.