A $n$-dimensional tangle is a collection $\mathcal{C}$ of properly embedded $n-2$-disks in a $n$-disk $D^n$. A tangle is said trivial if all the disks in $\mathcal{C}$ are simultaneously ambient isotopic into $\partial D^n$.
I'd like to prove that a tangle is trivial if there exists a radial Morse function $h:\text{int} (D^n)\to [0,\infty)$ (i.e. a Morse function with only a critical point of index 0 in the center of $D^n$) such that $h_{|\mathcal{C}}$ is a Morse function that only has index 0 critical points, one on each component of $\mathcal{C}$.
I'm only interested on the cases $n=3,4$, but maybe this can be proved in general.
This is my attempt for $n=3$: the hypothesis tells us that each arc only has a maximum point and no minima. We can drag all the maxima to the same height so that the tangle appears as a braid closed on the top as in the following diagram. Then we can find an isotopy od $D^n$ which bring the braid into the trivial one. We can think of moving the points of $\mathcal{C}\cap \partial D^n$ in order to remove each crossing in the braid, from bottom to top. Since the property of being a trivial tangle is preserved by global isotopies of $D^n$, we are done.
