David Futer

The Jones polynomial and graphs on surfaces

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.

On canonical triangulations of once-punctured torus bundles and two-bridge link complements

We prove the hyperbolization theorem for punctured-torus bun- dles and two-bridge links by decomposing them into ideal tetrahedra which are then given hyperbolic structures.

Guts of surfaces and the colored Jones polynomial

1 Introduction.- 2 Decomposition into 3-balls.- 3 Ideal Polyhedra.- 4 I-bundles and essential product disks.- 5 Guts and fibers.- 6 Recognizing essential product disks.- 7 Diagrams without non-prime arcs.- 8 Montesinos links.- 9 Applications.- 10 Discussion and questions.

ALTERNATING SUM FORMULAE FOR THE DETERMINANT AND OTHER LINK INVARIANTS

A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link. We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link. Furthermore, we obtain formulas for coefficients of the Jones polynomial by counting quantities on dessins. In particular, we will show that the jth coefficient of the Jones polynomial is given by sub-dessins of genus less or equal to j.

Links with no exceptional surgeries

We show that if a knot admits a prime, twist-reduced diagram with at least 4 twist regions and at least 6 crossings per twist region, then every non-trivial Dehn filling of that knot is hyperbolike. A similar statement holds for links. We prove this using two arguments, one geometric and one combinatorial. The combinatorial argument further implies that every link with at least 2 twist regions and at least 6 crossings per twist region is hyperbolic and gives a lower bound for the genus of a link.

SLOPES AND COLORED JONES POLYNOMIALS OF ADEQUATE KNOTS

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.

Cusp Areas of Farey Manifolds and Applications to Knot Theory

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured-torus bundles, 4-punctured sphere bundles, and two-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tessellation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example, sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial and the volume of hyperbolic knots, and to disprove a related conjecture.

Symmetric links and conway sums. Volume and jones polynomial

We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.

Angled decompositions of arborescent link complements

This paper describes a way to subdivide a 3-manifold into angled blocks, namely polyhedral pieces that need not be simply connected. When the individual blocks carry dihedral angles that fit together in a consistent fashion, we prove that a manifold constructed from these blocks must be hyperbolic. The main application is a new proof of a classical, unpublished theorem of Bonahon and Siebenmann: that all arborescent links, except for three simple families of exceptions, have hyperbolic complements. In the 1990s, Andrew Casson introduced a powerful technique for constructing and studying cusped hyperbolic 3-manifolds. His idea was to subdivide a manifold M into angled ideal tetrahedra: that is, tetrahedra whose vertices are removed and whose edges carry prescribed dihedral angles. When the dihedral angles of the tetrahedra add up to 2! around each edge of M, the triangulation is called an angled triangulation. Casson proved that every orientable cusped 3-manifold that admits an angled triangulation must also admit a hyperbolic metric, and outlined a possible way to find the hyperbolic metric by studying the volumes of angled tetrahedra — an idea also developed by Rivin [17]. The power of Casson’s approach lies in the fact that the defining equations of an angled triangulation are both linear and local, making angled triangulations relatively easy to find and deform (much easier than to study an actual hyperbolic triangulation, as in [14, 20] or in some aspects of Thurston’s seminal approach [22]). The aim of this paper is to extend this approach to larger and more complicated building blocks. These blocks can be ideal polyhedra instead of tetrahedra, but they may also have nontrivial topology. In general, an angled block will be a 3-manifold whose boundary is subdivided into faces looking locally like the faces of an ideal polyhedron (in a sense to be defined). The edges between adjacent faces carry prescribed dihedral angles. In Section 2, we will describe the precise combinatorial conditions that the dihedral angles must satisfy. These conditions will imply the following generalization of a result by Lackenby [11, Corollary 4.6].

Quasifuchsian state surfaces

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.