  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.
