We discuss a semigroup theoretic solution
to the problem of deciding when two braids
are equivalent under continuous deformations
where specified strings are allowed to pass
through each other, or merge and part. The
appropriate setting is within a general theory
of factorizable inverse monoids, in which the
coset monoid of a group plays a role, which is
dual to the classical theory of fundamental
inverse semigroups due to Munn. This is joint
work with Des FitzGerald and James East.