Random Operator Approach for Word Enumeration in Braid Groups

TL;DR

This paper introduces a symbolic dynamics method to analytically enumerate primitive words in braid groups, revealing that long primitive word counts are robust against local relations, with connections to random operator theory and modular functions.

Contribution

It develops a novel symbolic dynamics approach for exact enumeration in locally free groups and supports conjectures about the invariance of primitive word counts in braid groups.

Findings

Long primitive words are insensitive to local commutation relations.

The symbolic dynamics method enables exact enumeration.

Connections established with random operator theory and modular functions.

Abstract

We investigate analytically the problem of enumeration of nonequivalent primitive words in the braid group B_n for n >> 1 by analysing the random word statistics and the target space on the basis of the locally free group approximation. We develop a "symbolic dynamics" method for exact word enumeration in locally free groups and bring arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations. We consider the connection of these problems with the conventional random operator theory, localization phenomena and statistics of systems with quenched disorder. Also we discuss the relation of the particular problems of random operator theory to the theory of modular functions