Is the difference of two recursively enumerable sets, reducible to $K$? Is the difference of two recursively enumerable sets, reducible to $K$?
$W_x/W_y=\{z|z \in W_x  \&  z \notin W_y\}$
$K=\{x|\Phi_x(x) \downarrow\}$
$W_x= \text{dom}(\Phi_x)$

Which kind of reduction are you looking for?
It is trivially Turing reducible, because if you have a $K$-oracle available you can simply decide $W_x$ and $W_y$.
It's not many-one reducible, however, because it you take $W_x$ to be everything, a many-one reduction would tell you that the complement of a r.e. language is r.e. (which is well known to be false).
