  Given a 2-stranded tangle in a $\ZZ/2$ homology ball, $T\subset Y$, we
investigate the character variety $R(Y,T)$ of conjugacy classes of traceless
SU(2) representations of $\pi_1(Y\setminus T)$. In particular we completely
determine the subspace of binary dihedral representations, and identify all of
$R(Y,T)$ for many tangles naturally associated to knots in S^3. Moreover, we
determine the image of the restriction map from $R(T,Y)$ to the traceless SU(2)
character variety of the 4-punctured 2-sphere (the {\it pillowcase}). We give
examples to show this image can be non-linear in general, and show it is linear
for tangles associated to pretzel knots.
