Generating random braids

TL;DR

This paper introduces a polynomial-time algorithm for generating positive braids uniformly at random and describes an automaton for minimal lexicographic representatives, highlighting complexity and automaton state bounds.

Contribution

It presents a novel polynomial-time algorithm for random braid generation and characterizes the automaton accepting minimal lexicographic braid representatives.

Findings

Algorithm generates positive braids uniformly at random.

Automaton for minimal lexicographic representatives has exponential states.

Complexity is polynomial in strands and length.

Abstract

We present an algorithm to generate positive braids of a given length as words in Artin generators with a uniform probability. The complexity of this algorithm is polynomial in the number of strands and in the length of the generated braids. As a byproduct, we describe a finite state automaton accepting the language of lexicographically minimal representatives of positive braids that has the minimal possible number of states, and we prove that its number of states is exponential in the number of strands.