A combinatorial genesis of the right-angled relations in Artin's classical braid groups

Abstract

The configuration space $\text{UC}(n,p\times q)$ of $n$ unlabelled non-overlapping unit squares in a $p\times q$ rectangle is known to recover the homotopy type of the classical configuration space of $n$ unlabelled points in the plane, provided $\min\{p,q\}\geq n$. Thus the fundamental group $B_n(p\times q)$ of $\text{UC}(n,p\times q)$ yields a $(p,q)$-approximation of Artin's classical braid group $B_n$. We describe a right-angled Artin group presentation for $B_n(p\times q)$ in cases where $\text{UC}(n,p\times q)$ is known to be aspherical. When $\min\{p,q\}=2$, our presentation agrees with Artin's classical presentation for $B_n$ removing the Artin-Tits relations. This allows us to deduce the value of the Lusternik-Schnirelmann category of the corresponding aspherical spaces $\text{UC}(n,p\times q)$, as well as the values of all their $k$-sequential topological complexities, both in the classical (Rudyak et al.) and distributional (Dransihnikov et al.) contexts.