Quasifuchsian state surfaces

Abstract

This paper continues our study 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 A – or B B –adequate diagrams, the geometric type of the surface is also completely determined by a coefficient of the colored Jones polynomial of the link.