  We construct subfactors where one of the principal graphs is a spoke graph
using an algorithm which computes two-strand jellyfish relations. One of the
subfactors we construct is a 3^{Z/4} subfactor known to Izumi, which has not
previously appeared in the literature. To do so, we provide a systematic
treatment of the space of second annular consequences, which is analogous to
Jones' treatment of the space of first annular consequences in his quadratic
tangles article.
  This article is the natural followup to two recent articles on spoke
subfactor planar algebras and the jellyfish algorithm. Work of Bigelow-Penneys
explains the connection between spoke subfactor planar algebras and the
jellyfish algorithm, and work of Morrison-Penneys automates the construction of
subfactors where both principal graphs are spoke graphs using one-strand
jellyfish.
