  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 $n>3$.
