Relations amongst twists along Montesinos twins in the 4-sphere

Isotopy classes of diffeomorphisms of the 4-sphere can be described either from a Cerf theoretic perspective in terms of loops of 5-dimensional handle attaching data, starting and ending with handles in cancelling position, or via certain twists along submanifolds analogous to Dehn twists in dimension two. The subgroup of the smooth mapping class group of the 4-sphere coming from loops of 5-dimensional handles of index 1 and 2 coincides with the subgroup generated by twists along Montesinos twins (pairs of 2-spheres intersecting transversely twice) in which one of the two 2-spheres in the twin is unknotted. In this paper we show that this subgroup is in fact trivial or cyclic of order two.