Blocked-braid Groups

TL;DR

The paper introduces blocked-braid groups, a new family of groups derived from braid groups with geometric modifications, exploring their algebraic properties and connections to Frobenius algebras.

Contribution

It defines the blocked-braid groups, analyzes their structure, and establishes their finiteness for small n and infiniteness for larger n.

Findings

Blocked-braid groups are finite for n=1,2,3.

Blocked-braid groups are infinite for n>3.

Dirac's Belt Trick holds in these groups.

Abstract

We introduce and study a family of groups $\mathbf{BB}_n$, called the blocked-braid groups, which are quotients of Artin's braid groups $\mathbf{B}_n$, and have the corresponding symmetric groups $\Sigma_n$ as quotients. They are defined by adding a certain class of geometrical modifications to braids. They arise in the study of commutative Frobenius algebras and tangle algebras in braided strict monoidal categories. A fundamental equation true in $\mathbf{BB}_n$ is Dirac's Belt Trick; that torsion through $4\pi$ is equal to the identity. We show that $\mathbf{BB}_n$ is finite for $n=1,2$ and 3 but infinite for $n>3$.