Abstract: Every Kauffman state σ of a link diagram D(K) naturally defines a state surface S_σ whose boundary is K. For a homogeneous state σ, we show that K is a fibered link with fiber surface S_σ if and only if an associated graph G'_σ is a tree. As a corollary, it follows that for an adequate knot or link, the second and next-to-last coefficients of the Jones polynomial are obstructions to certain state surfaces being fibers for K.

This generalizes a theorem from paper [19], with a dramatically simpler proof.

Abstract: This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [13] and [19], while this survey focuses on the main ideas and examples.

Abstract: Let F be a surface and suppose that φ : F → F is a pseudo-Anosov homeomorphism fixing a puncture p of F. The mapping torus M = M_φ is hyperbolic and contains a maximal cusp C about the puncture p.

We show that the area and height of the cusp torus ∂C are equal, up to explicit multiplicative error, to the stable translation distance of φ acting on the arc complex A(F,p). Our proofs rely on elementary facts about the hyperbolic geometry of pleated surfaces. In particular, we do not use any deep results in Teichmüller theory, Kleinian group theory, or the coarse geometry of A(F,p).

A similar result holds for quasi-Fuchsian manifolds N ≈ F × R. In that setting, we prove a combinatorial estimate on the area and height of the cusp annulus in the convex core of N and give explicit multiplicative and additive errors.

Abstract: This work derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A-adequacy), we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and only if a particular coefficient vanishes. Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4.

Our approach is to generalize the checkerboard decompositions of alternating knots. Under mild diagrammatic hypotheses (A-adequacy), we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We employ normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). Since state graphs have previously appeared in the study of Jones polynomials, our setting and methods create a bridge between quantum and geometric knot invariants.

Abstract: Let M be a one-cusped hyperbolic manifold, and τ an unknotting tunnel for M. In the case where M is obtained by "generic" Dehn filling on one cusp of a two-cusped hyperbolic manifold, we prove that τ is isotopic to a geodesic, and characterize whether τ is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of τ relative to a maximal cusp. These results give generic answers to three long-standing questions posed by Adams, Sakuma, and Weeks.

We also construct an explicit sequence of one-tunnel knots in S³, all of whose unknotting tunnels have length approaching infinity.

Abstract: Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubinstein, Segerman, and Tillmann. Our construction leads to explicit lower bounds on the smallest angle in this positive angle structure, and to information about angled holonomy of the boundary tori.

Abstract: Let I_p,v be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K_v,v. We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(I_p,v) such that the quotient of I_p,v by Γ is a compact orientable surface of genus g. Surprisingly, the existence of Γ depends upon the value of v. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of Γ, together with a theorem of Haglund, implies that for p >= 6, every uniform lattice in Aut(I_p,v) contains a surface subgroup. We use elementary group theory, combinatorics, algebraic topology, and number theory.

Abstract: Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links.

We study the geometry of twisted torus links and related generalizations. We determine upper bounds on their hyperbolic volumes that depend only on the number of strands being twisted. We exhibit a family of twisted torus knots for which this upper bound is sharp, and another family with volumes approaching infinity. Consequently, we show there exist twisted torus knots with arbitrarily large braid index and yet bounded volume.

Abstract: This survey paper contains an elementary exposition of Casson and Rivin's technique for finding the hyperbolic metric on a 3-manifold M with toroidal boundary. We also survey a number of applications of this technique.

The method involves subdividing M into ideal tetrahedra and solving a system of gluing equations to find hyperbolic shapes for the tetrahedra. The gluing equations decompose into a linear and non-linear part. The solutions to the linear equations form a convex polytope A. The solution to the non-linear part (unique if it exists) is a critical point of a certain volume functional on this polytope. The main contribution of this paper is an elementary proof of Rivin's theorem that a critical point of the volume functional on A produces a complete hyperbolic structure on M.

Abstract: 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.

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 criterion, 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 τ. This paper proves that K admits a strong involution that fixes tau pointwise if and only if K is a two-bridge knot and τ 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.