Use of meanders and train tracks for description of defects and textures in liquid crystals and 2+1 gravity

TL;DR

This paper introduces a novel approach using meanders and train tracks to analyze defects and textures in liquid crystals and extends the methodology to 2+1 gravity, providing a new perspective on phase analysis.

Contribution

It applies meanders and train tracks to describe defects in liquid crystals and proposes their use in analyzing 2+1 gravity, offering a new analytical framework.

Findings

Meanders and train tracks effectively describe defects and textures.

The approach suggests possible phases and phase transitions in liquid crystals and gravity.

A master equation for 2+1 gravity is derived using train tracks.

Abstract

In this work (PartI) the qualitative analysis of statics and dynamics of defects and textures in liquid crystals is performed with help of meanders and train tracks. It is argued that similar analysis can be applied to 2+1 gravity. More rigorous justifications are presentedin the companion paper (PartII). Meanders were recently introduced by V.Arnold (Siberian J.of Math. Vol.29,36(1988)). Train tracks were originally introduced by W.Thurston in 1979 in his Princeton U. Lecture Notes (http://www.msri.org/gt3m/) in connection with description of self-homeomorphisms of 2 dimensional surfaces. Using train tracks the master equation is obtained which could be used alternatively to the Wheeler-DeWitt equation for 2+1 gravity. Since solution of this equation requires a large scale numerical work, in this paper we resort to the approximation of train tracks by the meandritic labyrinths. This allows us to analyse possible phases (and phase transitions)in both liquid crystals and gravity using Peierls- like arguments.