Conformal Geometry and Invariants of 3-strand Brownian Braids

TL;DR

This paper introduces a conformal geometric method to construct topological invariants of 3-strand Brownian braids, enabling analysis of their complexity and statistical properties, especially for indistinguishable particles.

Contribution

It presents a novel approach using conformal maps to analyze topological invariants and complexity of 3-strand Brownian braids, including the expectation of braid length.

Findings

Computed the expectation value of irreducible braid length.

Analyzed statistical properties of topological complexity.

Focused on indistinguishable particles in Brownian braids.

Abstract

We propose a simple geometrical construction of topological invariants of 3-strand Brownian braids viewed as world lines of 3 particles performing independent Brownian motions in the complex plane z. Our construction is based on the properties of conformal maps of doubly-punctured plane z to the universal covering surface. The special attention is paid to the case of indistinguishable particles. Our method of conformal maps allows us to investigate the statistical properties of the topological complexity of a bunch of 3-strand Brownian braids and to compute the expectation value of the irreducible braid length in the non-Abelian case.