  We consider a class of topological objects in the 3-sphere $S^3$ which will
be called {\it $n$-punctured ball tangles}. Using the Kauffman bracket at
$A=e^{\pi i/4}$, an invariant for a special type of $n$-punctured ball tangles
is defined. The invariant $F$ takes values in $PM_{2\times2^n}(\mathbb Z)$,
that is the set of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar
multiplication of $\pm1$. This invariant leads to a generalization of a theorem
of D. Krebes which gives a necessary condition for a given collection of
tangles to be embedded in a link in $S^3$ disjointly. We also address the
question of whether the invariant $F$ is surjective onto
$PM_{2\times2^n}(\mathbb Z)$. We will show that the invariant $F$ is surjective
when $n=0$. When $n=1$, $n$-punctured ball tangles will also be called
spherical tangles. We show that $\text{det} F(S)=0$ or 1 {\rm mod} 4 for every
spherical tangle $S$. Thus $F$ is not surjective when $n=1$.
