Non-integral Dehn surgeries characterize composite knots

A Dehn surgery slope p/q is said to be characterizing for a knot K if the homeomorphism type of the p/q-Dehn surgery along K determines the knot up to isotopy.
I show that if K is a composite knot, then every non-integral slope is characterizing for K. This yields the first and only examples to date of a complete and non-empty
list of non-characterizing slopes for a knot.