Growth Series of the Braid Monoid M B 5 in Band Generators

Abstract

Growth series is an important invariant associated with group or monoid which classifies all the words of group or monoid. Therefore, the growth series of braid monoids and Hecke algebras in Artin’s generators is presented in many scholarly published articles. The growth series of braid monoids $\text{M}{\text{B}}_3$ and $\text{M}{\text{B}}_4$ in band generators is known. In this work, we compute the complete presentation of braid monoid $\text{M}{\text{B}}_5$ in band generators by solving all the ambiguities of $\text{M}{\text{B}}_5$ . The words on the left-hand of each relation are reducible words, and the words on the right-hand side are canonical words. We partially find the growth series $\left(Q_{\ast }^{\left(5\right)}\right)$ of reducible words. Then, we construct a linear system for canonical words of $\text{M}{\text{B}}_5$ in band presentation and compute the corresponding growth series. We also find the growth rate of growth series of $\text{M}{\text{B}}_5$ in band generators.