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For a 4-manifold M and a knot k: S1↪∂M with dual sphere G: S2↪∂M, we compute the set D(M;k) of smooth isotopy classes of neat embeddings D2↪M with boundary k using an invariant going back to Dax. Moreover, we construct a group structure on D(M;k) and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group D(M;k) to the mapping class group of M.
Kosanović et al. (Fri,) studied this question.
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