# Difference between $MCG(ext(K))$ and $MCG(S^{3} - K)$

Can someone explain the difference between the mapping class groups of the knot complement and knot exterior?

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Are you worried about the difference between the mapping class group of a compact manifold with boundary and the mapping class group of its interior? –  Ian Agol Feb 18 '13 at 3:07
I don't know what your notation indicates precisely but by `exterior' do you mean $S^3$ with an open tubular neighbourhood of $K$ removed? If that's what you mean, then there's a restriction map from $MCG(ext(K))$ to $MCG(S^3 - K)$, and it's an isomorphism for all knots $K$. Or are you asking another question? But if by $MCG(ext(K))$ you demand the diffeomorphisms to be the identity on the boundary, then these groups are never isomorphic. –  Ryan Budney Feb 18 '13 at 3:11