  For a 4-manifold $M$ and a knot $k\colon\mathbb{S}^1\hookrightarrow\partial
M$ with dual sphere $G\colon\mathbb{S}^2\hookrightarrow\partial M$, we compute
the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings
$\mathbb{D}^2\hookrightarrow M$ with boundary $k$, using an invariant going
back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ 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 $\mathbb{D}(M;k)$ to the mapping class group of $M$.

Geometric Topology;Algebraic Topology