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Surgery on framed links give orientable compact closed connected 3-manifolds. Can you do surgery on framed tangles? Would Fenn-Rourke moves be invariant?

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It's not clear what you mean by surgery. Consider the case of a framed knot. Surgery involves the replacement of a D2×S1 by a complementary S1×D2, since the two spaces being exchanged have homeomorphic boundaries, namely S1×S1.

Now, if we try to do the same with a framed tangle, say with just 1 component, then we begin with an embedded D2×D1. The problem is that the second factor is not the boundary of any manifold, as it has boundary itself.

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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?".

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