  We study the cokernel of the Johnson homomorphism for the mapping class group
of a surface with one boundary component. A graphical trace map simultaneously
generalizing trace maps of Enomoto-Satoh and Conant-Kassabov-Vogtmann is given,
and using technology from the author's work with Kassabov and Vogtmann, this is
is shown to detect a large family of representations which vastly generalizes
series due to Morita and Enomoto-Satoh. The Enomoto-Satoh trace is the rank 1
part of the new trace. The rank 2 part is also investigated.
