Diffeomorphisms of 4-manifolds from graspers

We relate degree one grasper families of embedded circles to various constructions of diffeomorphisms found in the literature – theta clasper classes of Watanabe, barbell diffeomorphisms of Budney and Gabai, and twin twists of Gay and Hartman. We use a ``parameterised surgery map’’ that for a smooth 4-manifold $M$ takes loops of framed embeddings of $S^1$ in the manifold obtained by surgery on some 2-sphere in $M$, to the mapping class group of $M$.