TITLE: Can you do surgery on framed tangles?
QUESTION [6 upvotes]: Surgery on framed links give orientable compact closed connected 3-manifolds. Can you do surgery on framed tangles? Would Fenn-Rourke moves be invariant?

REPLY [3 votes]: You can cut out a collection of 3-balls (a regular neighbourhood of the tangle), and glue them back in a different way. Such modifications are a part of the Montesinos trick. Montesinos uses them to prove that any compact oriented connected 3-manifold is a 3-fold branched cover of the 3-sphere. In this case, the "surgery on the tangle" downstairs lifts to honest Dehn surgery upstairs.
In the context of branched covering spaces, your second question generalizes to the open question "does the Kirby theorem hold for surgery in an orbifold?".