A Geometric Approach to the Links-Quivers Correspondence I: Rational Tangles

TL;DR

This paper introduces a geometric method to establish the Links-Quivers Correspondence specifically for rational tangles, enabling the recovery of colored HOMFLY-PT polynomials from associated quivers.

Contribution

It provides a new geometric proof for the Links-Quivers Correspondence tailored to rational tangles and explicitly describes the quivers using winding numbers in the punctured plane.

Findings

Explicit quiver descriptions for rational tangles

Geometric proof of the Links-Quivers Correspondence

Connection between winding numbers and quiver data

Abstract

The Links-Quivers Correspondence predicts that all the symmetric (or antisymmetric) colored HOMFLY-PT polynomials of a link can be recovered from a finite amount of data (a quiver) associated to the link. We give a new geometric proof of the Links-Quivers Correspondence modified for rational tangles and explicitly describe the corresponding quivers in terms of winding numbers in the punctured plane and its second configuration space.