arXiv research

Study of 3-manifold census manifolds and compare with random models.

We give a short proof of Masbaum and Reid's result that mapping class groups involve any finite group, appealing to free quotients of surface groups and a result of Gilman, following Dunfield-Thurston.

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfie…

New maps connect universal circles to ideal sphere for hyperbolic manifolds.

Law of large numbers for volumes of random 3-manifolds.

Study shows universal circles for taut foliations in 3-manifolds.

Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.

This paper characterizes Fuchsian groups acting on the circle with invariant laminations.

The paper proves inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.

The paper explores universal circles for Anosov foliations and their uniqueness.

Standardizes Dunfield-Gong's 4-sphere, solves knot sliceness problem.

We study random elements of subgroups (and cosets) of the mapping class group of a closed hyperbolic surface, in part through the properties of their mapping tori. In particular, we study the distribution of the homology of the mapping torus (with rational, integer, and finite field coefficients, the hyperbolic volume …

Study shows universal circle isomorphic to flow space ideal boundary.

We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first eigenvalue of the laplacian on coexact one-forms is bounded above explicitly in te…

We propose a program to study groups acting faithfully on S^1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups i…

In this short note, we show that the twisted Alexander polynomial associated to a parabolic SL(2,C)-representation detects genus and fibering of the twist knots. As a corollary, a conjecture of Dunfield, Friedl and Jackson is proved for the hyperbolic twist knots.

The study explores finite quotients of 3-manifold groups and their existence and non-existence.

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the $n$th colored Jones polynomial at $e^{\a/n}$, when $\a$ is a fixed complex number and $n$ tends to infinity. We analy…

We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational. Dunfield's proof involves the rigidity of maximal volume. The volume is still an important ingredient in this paper. Unfortunately at this poi…

These are the extended notes of a talk I gave at the Geometric Topology Seminar of the Max Planck Institute for Mathematics in Bonn on January 30th, 2012. My goal was to familiarize the topologists with the basics of arithmetic hyperbolic 3-manifolds and sketch some interesting results in the theory of 3-manifolds (suc…

Character varieties of prime knots have high-dimensional components.

We improve upon a recent result of Culler and Dunfield on orderability of certain Dehn fillings by removing a difficult condition they required.

We propose an algebraic model of the conjectural triply graded homology of Gukov, Dunfield and Rasmussen for some torus knots. It turns out to be related to the q,t-Catalan numbers of Garsia and Haiman.

In this paper we show that the twisted Alexander polynomial associated to a parabolic representation determines fiberedness and genus of a wide class of 2-bridge knots. As a corollary we give an affirmative answer to a conjecture of Dunfield, Friedl and Jackson for infinitely many hyperbolic knots.

We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy represe…

We show the problem of counting homomorphisms from the fundamental group of a homology $3$-sphere $M$ to a finite, non-abelian simple group $G$ is #P-complete, in the case that $G$ is fixed and $M$ is the computational input. Similarly, deciding if there is a non-trivial homomorphism is NP-complete. In both reductions,…

The paper constructs new asymmetric hyperbolic manifolds with lens space fillings.

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in $S^{3}$ is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-po…

The coefficients of twisted Alexander polynomials of a knot induce regular functions of the $SL_2(\mathbb{C})$-character variety. We prove that the function of the highest degree has a finite value at an ideal point which gives a minimal genus Seifert surface by Culler-Shalen theory. It implies a partial affirmative an…

In this paper we apply the twisted Alexander polynomial to study the fibering and genus detecting problems for oriented links. In particular we generalize a conjecture of Dunfield, Friedl and Jackson on the torsion polynomial of hyperbolic knots to hyperbolic links, and confirm it for an infinite family of hyperbolic 2…

New operators in Khovanov-Rozansky homology exhibit symmetry.

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map from a Dehn surgery component in the PSL(2,C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step …

This paper proves a conjecture about knot homologies.

In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient …

Survey on Thurston metric on Teichmüller space, focusing on extremal maps.

Unified proof of Nielsen-Thurston classification via Teichmüller's theorem.

The paper extends Thurston's method to new variants of Mather-Thurston theorem.

Developed theory for Thurston maps with a small set of essential singularities.

Article explores Thurston's circle packing theorem in 3-manifold geometry.

Overview of Thurston's work in math.

We establish basic geometric and topological properties of Thurston's Master Teapot and the Thurston set for superattracting unimodal self-maps of intervals. In particular, the Master Teapot is connected, contains the unit cylinder, and its intersection with a set $\mathbb{D} \times \{c\}$ grows monotonically with $c$.…

Study shows genus two Heegaard diagrams for all Thurston geometries.

Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, incl…

Uniformly finite Cannon--Thurston fibers in most hyperbolic settings.

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This comple…

We show that Cannon-Thurston maps exist for degenerate free groups without parabolics, i.e. for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon-Thurston maps for surface groups, we show that Cannon-Thurston maps exist for arbitrary finitely generated Kleinian groups witho…

The Slope Conjecture relates a quantum knot invariant, (the degree of the colored Jones polynomial of a knot) with a classical one (boundary slopes of incompressible surfaces in the knot complement). The degree of the colored Jones polynomial can be computed by a suitable (almost tight) state sum and the solution of a …

Thurston's jiggling lemma simplifies triangulations.