  We provide a direct connection between the $\mathcal{Z}_{\max}$ (or
essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic
two-generator one-relator group with abelianisation of rank $2$.
  We deduce various structural and algorithmic properties, like the existence
of a quadratic-time algorithm computing the $\mathcal{Z}_{\max}$-JSJ
decomposition of such groups.
