Abstract: In recent years, several families of hyperbolic knots have been shown to have both volume and λ₁ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized twist number of a diagram can provide two-sided bounds on either the volume or λ₁. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ₁. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded.

Abstract: We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Earlier work by Mattman, combined with Agol and Lackenby's 6-theorem, reduces the argument to knots with small indices p,q,r. We treat these using the Culler-Shalen norm of the SL(2,C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety.

Abstract: We find explicit, combinatorial estimates for the cusp areas of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. Applications include volume estimates for the hyperbolic 3-manifolds obtained by Dehn filling these bundles, for example estimates on the volume of closed 3-braid complements in terms of the complexity of the braid word. We also relate the volume of a closed 3-braid to certain coefficients of its Jones polynomial.

Abstract: 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.

Abstract: 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 other link invariants by counting quantities on dessins.

Abstract: 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 plolynomial of planar graphs to graphs that are embedded in closed surfaces of higher genus (i.e. dessins d'enfant).

In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain dessin associated to a link projection. We give some applications of this approach.

Abstract: 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.

Abstract: Given a hyperbolic 3-manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.

Abstract: The complements of two-bridge links in S³ have a natural decomposition into topological ideal tetrahedra, described by Sakuma and Weeks. Following the lead of Guéritaud, we use volume maximization techniques to give this ideal triangulation a complete hyperbolic structure. Applications of this method include sharp volume estimates and a result (conjectured by Thistlethwaite) about arcs in the projection plane being hyperbolic geodesics.

Abstract: If Thurston's Geometrization Conjecture is true, then a closed 3-manifold is hyperbolic whenever it satisfies a topological condition, called "hyperbolike". This paper proves a mild diagrammatic condition on a knot or link in S³ under which any non-trivial Dehn filling gives a hyperbolike closed manifold. For a knot K, a non-trivial Dehn filling of K will be hyperbolike whenever a prime, twist-reduced diagram of K has at least 4 twist regions and at least 6 crossings per twist region; the statement for links is similar.

We prove this result using two arguments, one geometric and one combinatorial. The combinatorial argument also proves 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.

Abstract: Let K be a knot that has an unknotting tunnel tau. This paper proves that K admits a strong involution that fixes tau pointwise if and only if K is a two-bridge knot and tau its upper or lower tunnel. One result obtained along the way is a version of the Smith conjecture for handlebodies: the fixed-point set of an orientation-preserving, periodic diffeomorphism of a handlebody is either empty or boundary-parallel.

Abstract: We model interfaces between immiscible fluids as cost-minimizing networks, where "cost" is a weighted length. We consider conjectured necessary and sufficient conditions for when a planar cone is minimizing. In some cases we give a proof; in other cases we provide a counterexample.