  Given a $d$-dimensional manifold $M$ and a knotted sphere
$s\colon\mathbb{S}^{k-1}\hookrightarrow\partial M$ with $1\leq k\leq d$, for
which there exists a framed dual sphere
$G\colon\mathbb{S}^{d-k}\hookrightarrow\partial M$, we show that the space of
neat embeddings $\mathbb{D}^k\hookrightarrow M$ with boundary $s$ can be
delooped by the space of neatly embedded $(k-1)$-disks, with a normal vector
field, in the $d$-manifold obtained from $M$ by attaching a handle to $G$. This
increase in codimension significantly simplifies the homotopy type of such
embedding spaces, and is of interest also in low-dimensional topology. In
particular, we apply the work of Dax to describe the first interesting homotopy
group of these embedding spaces, in degree $d-2k$. In a separate paper we use
this to give a complete isotopy classification of 2-disks in a 4-manifold with
such a boundary dual.
