Quasifuchsian state surfaces

TL;DR

This paper investigates essential state surfaces in link complements, establishing that their geometric type is determined by graph-theoretic criteria and link invariants like the colored Jones polynomial, especially for hyperbolic and adequate links.

Contribution

It provides a complete characterization of the geometric type of state surfaces in hyperbolic links using graph criteria and polynomial invariants, extending previous work.

Findings

Geometric type determined by graph-theoretic criterion for hyperbolic links.

For adequate links, the colored Jones polynomial coefficient determines surface type.

Extends understanding of the relationship between link invariants and geometric structures.

Abstract

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.