  Watanabe's theta graph diffeomorphism, constructed using Watanabe's clasper
surgery construction which turns trivalent graphs in 4-manifolds into
parameterized families of diffeomorphisms of 4-manifolds, is a diffeomorphism
of $S^4$ representing a potentially nontrivial smooth mapping class of $S^4$.
The "(1,2)-subgroup" of the smooth mapping class group of $S^4$ is the subgroup
represented by diffeomorphisms which are pseudoisotopic to the identity via a
Cerf family with only index 1 and 2 critical points. This author and Hartman
showed that this subgroup is either trivial or has order 2 and explicitly
identified a diffeomorphism that would represent the nontrivial element if this
subgroup is nontrivial. Here we show that the theta graph diffeomorphism is
isotopic to this one possibly nontrivial element of the (1,2)-subgroup. To
prove this relation we develop a diagrammatic calculus for working in the
smooth mapping class group of $S^4$.
