Suppose I have a 3 bridge 3 component link. I know immediately that my components are unknots. I want to know if there is a more simple way to understand these, though. Draw your picture as follows: start with three bridges on top, insert an appropriate element of the braid group of 6 strands, connect the ends of the braid to 3 upside-down bridges. We had to be careful with our choice of the braid, since imposing that we connect the braid at the bottom in this way introduces the possibility that we don't have 3 components.
Given this setup, is there an easy way to see this "component preserving" subgroup of the braid group?
Obviously this does not restrict to 3 strands, one can talk about n bridge n component, but I want to look at 3 for a problem I am working on.