Knots/links seem to be studied quite a lot for their topological connection to 3-manifolds by considering knot complements in $S^{3}$. Is there an analogous topological entity for braids? They appear too similar for there not to be one.
2026-03-28 13:42:12.1774705332
Analogous notion of knot complements for braids
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Definitely. You can think of a braid as an embedding of a union of intervals into $D^2\times I$ where the ends go to prescribed points and the strands of the embedding don't "backtrack." (If you allow backtracking you get what are called string links.) So, if $B$ is a braid, you could look at the complement $D^2\times I\setminus B$, which can tell you a lot about the link.