Abstract: We give a formula to estimate the number of fixed points of a pseudo-Anosov homeomorphism of a surface. When the homeomorphism satisfies a mild property called strong irreducibility, the log of the number of fixed points is coarsely equal to the Teichmuller translation length. We also discuss several applications, including an inequality relating the hyperbolic volume of a mapping torus to the rank of its Heegaard Floer homology.

Abstract: This paper proves that every periodic automorphism of a closed hyperbolic surface $S$ sends some curve to a nearly disjoint curve. In particular, periodic maps cannot have the property that every curve fills with its image, so no such map can give a positive answer to a question of Wright. Finally, this paper answers a question of Schleimer about irreducible periodic surface maps.

Abstract: The family of right-angled tiling links consists of links built from regular 4-valent tilings of constant-curvature surfaces that contain one or two types of tiles. The complements of these links admit complete hyperbolic structures and contain two totally geodesic checkerboard surfaces that meet at right angles. In this paper, we give a complete characterization of which right-angled tiling links are arithmetic, and which are pairwise commensurable. The arithmeticity classification exploits symmetry arguments and the combinatorial geometry of Coxeter polyhedra. The commensurability classification relies on identifying the canonical decompositions of the link complements, in addition to number-theoretic data from invariant trace fields.

Abstract: This paper employs knot invariants and results from hyperbolic geometry to develop a practical procedure for checking the cosmetic surgery conjecture on any given one-cusped manifold. This procedure has been used to establish the following computational results. First, we verify that all knots up to 19 crossings, and all one-cusped 3-manifolds in the SnapPy census, do not admit any purely cosmetic surgeries. Second, we check that a hyperbolic knot with at most 15 crossings only admits chirally cosmetic surgeries when the knot itself is amphicheiral. Third, we enumerate all knots up to 13 crossings that share a common Dehn fillings with the figure-8 knot. The code that verifies these results is publicly available on GitHub.

Abstract: A finite-dimensional CAT(0) cube complex \(X\) is equipped with several well-studied boundaries. These include the Tits boundary (which depends on the CAT(0) metric), the Roller boundary (which depends only on the combinatorial structure), and the simplicial boundary (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of the Roller boundary to define the simplicial Roller boundary. Then, we show that the Tits boundary, the simplicial Roller boundary, and teh simplicial boundary are all homotopy equivalent in a way that is \(Aut(X)\)-equivariant up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition that provides a dictionary among several distinct perspectives on cube complexes.

Abstract: We show that, for hyperbolic fibered knots in the three-sphere, the volume and the genus are unrelated. That is, we construct hyperbolic fibred knots with fixed genus and arbitrarily large volume.

Abstract: We show that low-density random quotients of cubulated hyperbolic groups are again cubulated and hyperbolic. Ingredients of the proof include cubical small-cancellation theory, the exponential growth of conjugacy classes, and the statement that hyperplane stabilizers grow exponentially more slowly than the ambient cubical group.

Abstract: We prove that every cusped hyperbolic 3-manifold has a finite cover admitting infinitely many geometric ideal triangulations. Furthermore, every long Dehn filling of one cusp in this cover admits infinitely many geometric ideal triangulations. This cover is constructed in several stages, using results about separability of peripheral subgroups and their double cosets, in addition to a new conjugacy separability theorem that may be of independent interest. The infinite sequence of geometric triangulations is supported in a geometric submanifold associated to one cusp, and can be organized into an infinite trivalent tree of Pachner moves.

Abstract: We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into theorems about infinite-volume manifolds. We also prove and apply an infinite-volume version of the 6-Theorem.

Abstract: This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold \(N\) and the thick part of any of its long Dehn fillings. Given a bilipschitz constant \(J\) and a thickness constant \(\varepsilon > 0\), we quantify how long a Dehn filling suffices to guarantee a \(J\)-bilipschitz map on \(\varepsilon\)-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation.

Our quantitative results have two immediate applications. First, we relate the Margulis number of $N$ to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed 3-manifold $M$ whose Margulis number is less than 0.29. Combined with Shalen's upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of 3-manifolds whose Margulis numbers are below 0.29.

Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold $N$, we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on $N$. This reduces the cosmetic surgery conjecture on $N$ to an explicit finite search.

Abstract: Every pseudo-Anosov mapping class $\varphi$ defines an associated veering triangulation $\tau_\varphi$ of a punctured mapping torus. We show that generically, $\tau_\varphi$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmuller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.

Abstract: We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.

Abstract: We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. We focus on determining when a link is hyperbolic, estimating its volume, and bounding its cusp shape and cusp area. We give sample applications and state some open questions and conjectures.

Abstract: This paper proves that every finite volume hyperbolic 3-manifold $M$ contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a crucial way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary of this result and a companion statement about surfaces with cusps, we recover Wise's theorem that the fundamental group of $M$ acts freely and cocompactly on a CAT(0) cube complex.

Abstract: This paper proves that non-elementary hyperbolic groups grow exponentially more quickly than their infinite index quasiconvex subgroups. The main tools are automatic structures and Perron-Frobenius theory.

We also extend the main result to relatively hyperbolic groups and cubulated groups. These extensions use the notion of growth tightness and the work of Dahmani, Guirardel, and Osin on rotating families.

Abstract: Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3-manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for all sufficiently large $n$, we construct a pair of incommensurable hyperbolic 3-manifolds $N_n$ and $N_n^\mu$ whose volume is approximately n and whose length spectra agree up to length $n$.

Both $N_n$ and $N_n^\mu$ are built by gluing two standard submanifolds along a complicated pseudo-Anosov map, ensuring that these manifolds have a very thick collar about an essential surface. The two gluing maps differ by a hyper-elliptic involution along this surface. Our proof also involves a new commensurability criterion based on pairs of pants.

Abstract: For each natural number $n \geq 4$, we determine the unique lowest volume hyperbolic 3-orbifold whose torsion orders are bounded below by $n$. This lowest volume orbifold has base space the 3-sphere and singular locus the figure-8 knot, marked $n$. We apply this result to give sharp lower bounds on the volume of a hyperbolic manifold in terms of the order of elements in its symmetry group.

Abstract: We provide a diagrammatic criterion for semi-adequate links to be hyperbolic. We also give a conjectural description of the satellite structures of semi-adequate links. One application of our result is that the closures of sufficiently complicated positive braids are hyperbolic links.

Abstract: We prove that in the complement of a highly twisted link, all closed, essential, meridionally incompressible surfaces must have high genus. The genus bound is proportional to the number of crossings per twist region. A similar result holds for surfaces with meridional boundary: such a surface either has large negative Euler characteristic, or is an $n$-punctured sphere visible in the diagram.

Abstract: This paper continues our study, initiated in the monograph [20], 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.

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

We show that the area and height of the cusp torus $\partial C$ are equal, up to explicit multiplicative error, to the stable translation distance of $\varphi$ acting on the arc complex $\mathcal{A}(F,p)$. 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 $\mathcal{A}(F,p)$.

A similar result holds for quasi-Fuchsian manifolds $N \cong F \times \mathbb{R}$. 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 paper proves lower bounds on the volume of a hyperbolic 3-orbifold whose singular locus is a link. We identify the unique smallest volume orbifold whose singular locus is a knot or link in the 3-sphere, or more generally in a $\mathbb{Z}_6$ homology sphere. We also prove more general lower bounds under mild homological hypotheses.

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 [20], while this survey focuses on the main ideas and examples.

Abstract: Every Kauffman state $\sigma$ of a link diagram $D(K)$ naturally defines a state surface $S_\sigma$ whose boundary is $K$. For a homogeneous state $\sigma$, we show that $K$ is a fibered link with fiber surface $S_\sigma$ if and only if an associated graph $G'_\sigma$ 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 provides a dramatically simpler proof of a theorem from monograph [20].

Abstract: This monograph 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: Any one-cusped hyperbolic manifold M with an unknotting tunnel $\tau$ is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that $\tau$ is isotopic to a geodesic, and characterize whether $\tau$ is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of $\tau$ 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 $S^3$, all of whose unknotting tunnels have length approaching infinity.

Abstract: We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic manifold $X$, and a Dehn filling whose meridian and longitude curves are longer than $2\pi(2g-1)$, we show that every genus $g$ Heegaard splitting of the filled manifold comes from a splitting of the original manifold $X$. The analogous statement holds for fillings of multiple boundary tori. This gives an effective version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.

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 $\mathcal{I}_{p,v}$ 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 $\Gamma$ in $Aut(\mathcal{I}_{p,v})$ such that the quotient of $\mathcal{I}_{p,v}$ by $\Gamma$ is a compact orientable surface of genus $g$. Surprisingly, the existence of $\Gamma$ depends upon the value of $v$. The remaining cases lead to open questions in tessellations of surfaces and in number theory. Our construction of $\Gamma$, together with a theorem of Haglund, implies that for $p \geq 6$, every uniform lattice in $Aut(\mathcal{I}_{p,v})$ 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 $\lambda_1$ (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 $\lambda_1$. 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 $\lambda_1$. 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 $SL(2,\mathbb{C})$-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 $2\pi$. 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 $S^3$ 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 $S^3$ 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.

Abstract: Frank Morgan has conjectured that the spherical caps in a standard double bubble satisfy a certain curvature inequality. These notes represent a partial proof of this conjecture. In 2006, Marilyn Daily proved the general case of the conjecture, relying on these notes for the 2-dimensional case.

Abstract: Given a prime, alternating link diagram, we build a special cover of the link complement whose degree is bounded by a factorial function of the crossing number. It follows that a subgroup of the link group of that index embeds into right-angled Artin and Coxeter groups. Corollaries of this result include a quantification of residual finiteness, control of the growth of Betti numbers in covers, and an explicit bound on the rank of a Z-module on which the link group acts faithfully.

This paper draft contains an error in Theorem 6.18. The statement of that theorem refers to a large commutative diagram of covering maps between graphs. While the commutative diagram is correct, the assertion that the graph of spaces in the top row covers the graph of spaces in the bottom row is mistaken due to a basepoint issue. Unfortunately, this mistake invalidates the proof of the main theorem.