Quasifuchsian state surfaces

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.Comment: 21 pages, 9 figures: To appear in the Transactions of the AM

A Jones slopes characterization of adequate knots

“…and we conclude that and −s * ≤ 2c − (D). Since by assumption s − s * = 2c(K), and c − (D) + c + (D) = c(K) we conclude that(11) s = 2c + (D) and − s * = 2c − (D).…”

“…Adequate knots form a large class of knots that behaves well with respect to Jones-type knot invariants and has nice topological and geometric properties [1,2,6,8,7,9,10,11,12,24,27]. Several well known classes of knots are adequate; these include all alternating knots and Conway sums of strongly alternating tangles.…”

An alternative approach to hyperbolic structures on link complements

“…Proof It is proved in [14] that the checkerboard surfaces for such link diagrams are incompressible and boundary incompressible, and it is proved in Adams [3], and Futer, Kalfagianni and Purcell [10] that they are quasi-Fuchsian, hence contain no accidental parabolics.…”

“…Note 1.3 In [10], the authors state their results for a more general class of diagrams than alternating. However, the spanning surfaces considered are so-called state surfaces; for non-alternating diagrams these are different from checkerboard surfaces, and we do not know at present whether these can be incorporated into our method of computing hyperbolic structures.…”

Crosscap numbers and the Jones polynomial

“…In the 80s Kauffman [27] and Murasugi [37] showed that the degree span of the Jones polynomial determines the crossing number of alternating links. More recently, Futer, Kalfagianni and Purcell showed that coefficients of the colored Jones polynomials contain information about incompressible surfaces in the link complement and have strong relations to geometric structures and in particular to hyperbolic geometry [17,15,16]. For instance, certain coefficients of the polynomials coarsely determine the volume of large classes of hyperbolic links [13,14], including hyperbolic alternating links as shown by Dasbach and Lin [11].…”