Complement of an unknotted oriented surface in $S^4$

Question

Let $H$ be a genus $g$ handlebody embedded in $S^4$ and let $X = S^4 - N(\partial H)$ where $N(\partial H)$ is an open tubular neighborhood of the boundary of $H$. What is $X$? In the case where $g=0$, for example, $X = S^1 \times D^3$.

Answer 1

To get a handle description for $S^4-\nu\Sigma$, push $\Sigma$ into one of the $D^4$'s and view $D^4$ as $I\times D^3$. Perturb $\Sigma$ so that the projection $I\times D^3\to I$ given by $(t,x)\mapsto t$ is a morse function with isolated critical values when restricted to $\Sigma$. The critical values of $t|\Sigma$ correspond to 2-dimensional handles in the handle decomposition for $\Sigma$. To get a handle description for the complement of the surface, we have to add a 4d $(k+1)-$handle for each 2d $k-$handle of $\Sigma$. The only tricky part is the 4d $2-$handles. Draw $\Sigma$ with a handle decomposition so that all $1-$handles are attached to $0-$handles. The $0-$handles of $\Sigma$ become dotted circles (4d $1-$handles) and each $1-$handle of $\Sigma$ (a band) is replaced with a $0-$framed circle so that on either side of the band, you push off the core of the band and connect these around the dotted circle.

So, the example above is a handlebody description for the complement of an unknotted torus. This is all from Gompf & Stipsicz, proposition 6.2.1 in conjunction with the neighboring discussion.