Quasifuchsian state surfaces

Abstract

This paper continues our study, initiated in [12], 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.

On the Degree of the Colored Jones Polynomial

“…In fact, Garoufalidis and Le have studied the "higher order" stability properties of the colored Jones polynomials and they showed that the stable coefficients of the polynomials J K (n + 1, q) give rise to infinitely many q-series with interesting number theoretic and physics connections. On the other hand, the work of Futer, Kalfagianni and Purcell [8,9,10,12,11] showed that certain stable coefficients of J K (n + 1, q) encode information about incompressible surfaces in knot complements and their geometric types and have direct connections to hyperbolic geometry. See also discussion below.…”

“…At each crossing of D, we connect the pair of neighboring disks by a half-twisted band to construct a surface S A ⊂ S 3 whose boundary is K. See Figure 1 for an example. By the work of the first author with Futer and Purcell [10,12], the invariant T K (q) detects the geometric types of the surface S A (D) and contains a lot of information about the geometric structures of of the complements S 3 \\S A (D) and S 3 K. For example, combining Corollary 3.6 with results of [10,12] we have the following; for terminology and more details the reader is referred to the original references.…”

“…The Jones polynomial and the colored Jones polynomials of semi-adequate links have been studied considerably in the literature [19,18,21] and [2,1,3,7,15,14,13] and they have been shown to capture deep information about incompressible surfaces and geometric structures of link complements [8,9,10,12,11].…”

“…We discuss how this invariant can be thought of as generalizing certain stable coefficients of the colored Jones polynomials of semi-adequate links, studied by Armond [1], Dasbach and Lin [7], and Garoufalidis and Le [13], to all links. We also discuss how, combined with work of Futer, Kalfagianni and Purcell [10,12], our invariant detects certain incompressible surfaces in link complements and their geometric types.…”

Quantum one-cocycles for knots

“…It seems that the main connections between them are the volume conjecture, which generalizes Kashaevs conjecture for hyperbolic knots (see [34] and [23]), the AJ-conjecture (see [18] and [32]) and the relations, via certain incompressible surfaces in a link complement, of the extremal coefficients of the n-colored Jones polynomial with the geometric structure of the link complement (see [15], [16], [17]).…”