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A slope r is characterising for a knot K if the diffeomorphism type of the manifold obtained by r-surgery along K uniquely determines K. In a joint work with Diego Santoro, we show that zero is a characterising slope for the untwisted Whitehead double of any knot K.
Although this is a purely 3-dimensional result, it has an interesting relation to 4-dimensional topology: an important open question asks whether a knot is slice if and only if its Whitehead double is slice.
Our theorem shows that the use of RBG-links to approach this question cannot work.
More generally, zero is a characterising slope for all n-twisted Whitehead doubles of K, with the assumption that n is not pq if K is a non-trivial (p,q)-torus knot or (p,q)-cable. In particular, zero is a characterising slope for all twist knots.