Consider a pure braid. I wonder whether there is a criterion that tells me when it splits?
What I mean here with "split": Let the top and bottom ends of the braid be attached to a disk, such that the bottom end of a given strand has the same coordinates as its bottom end. We can think of the braid running inside a cylinder = disc x interval. (cylinder black in the figure 1, strands red)
How do I know (from a braid word ideally) whether I find a disk whose boundary lies in the boundary of the cylinder, with two segments in the disk that follow the same path in the disk, and two segments running from the top to the (blue in the picture 1) that splits the cylinder into two parts in which each strands of the braid are contained?
This is not equivalent to the braid closure forming a split link.
If anyone has a reference or idea, that would be highly appreciated!
Thank you!
Edit: I want the segments of the splitting disk that run from top to bottom to really run vertically. For example, two braids that run around each other in the form of a helix would not admit such a seperating disk.[See middle figure]2
My "split braids" are also not the same as reducible braids if I understand the definition correctly (even if the splitting disk was allowed to have its boundary running around the cylinder): For example if the tubes of the reduced braid form the braid whose closure is the (3,1)-Torus link, then that should be a reduced braid but not split in the sense I am considering. [See right figure]2
That is, I am searching for a way to tell whether a pure braid is reducible with the tubes forming a trivial braid.