  This paper continues our study, initiated in [arXiv:1108.3370], of essential
state surfaces in link complements that satisfy a mild diagrammatic hypothesis
(homogeneously adequate). For hyperbolic links, we show that the geometric type
of these surfaces in the Thurston trichotomy is completely determined by a
simple graph--theoretic criterion in terms of a certain spine of the surfaces.
For links with A- or B-adequate diagrams, the geometric type of the surface is
also completely determined by a coefficient of the colored Jones polynomial of
the link.
