A new approach to light bulb tricks: Disks in 4-manifolds

For a 4-manifold $M$ and a knot $k\text{\hspace{0.17em}⁣}:{\mathbb{S}}^1\hookrightarrow \mathrm{\partial }M$ with dual sphere $G\text{\hspace{0.17em}⁣}:{\mathbb{S}}^2\hookrightarrow \mathrm{\partial }M$, 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 $M$.