Classification of Homogeneous Local Representations of the Singular Braid Monoid

For a natural number $n$, denote by $B_n$ the braid group on $n$ strings and by $S{M}_n$ the singular braid monoid on $n$ strings. $S{M}_n$ is one of the most important extensions of $B_n$. In [13], Y. Mikhalchishina classified all homogeneous $2$-local representations of $B_n$ for all $n\ge 3$. In this article, we extend the result of Mikhalchishina in two ways. First, we classify all homogeneous $3$-local representations of $B_n$ for all $n\ge 4$. Second, we classify all homogeneous $2$-local representations of $S{M}_n$ for all $n\ge 2$ and all homogeneous $3$-local representations of $S{M}_n$ for all $n\ge 4$.