Determining the filling factors of fractional quantum Hall states using knot theory

TL;DR

This paper introduces a topological knot theory-based method to determine filling factors in the fractional quantum Hall effect, leveraging Schubert's theorems to derive a new formula and suggesting a link to particle interactions and Berry phases.

Contribution

It proposes a novel approach using rational tangles and knot theory to calculate fractional quantum Hall filling factors, providing a new theoretical framework.

Findings

Derived a new formula for filling factors using knot theory

Connected knot isotopy to quantum Hall states

Suggested a link between particle interactions and Berry phases

Abstract

In this work a method based on a topological invariance of rational tangles commonly used in knot theory determines filling factors in the fractional quantum Hall effect. The main sustain for this hypothesis are the Schubert's theorems which treats the isotopic between two knots that are numerators of non-equivalent rational tangles. This isotopic allows to deduce a new formula for all filling factors. Besides, it opens a new perspective for a future connection between $N-$particles interaction at different fillings and Berry phase evaluated along torus knots