arXiv research

Classifies nonorientable surfaces in a specific type of bundle.

New CR structures found for once-punctured torus bundles.

The paper certifies projective rigidity for once-punctured torus bundles using twisted Alexander polynomials.

Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.

Paper finds generalized torsions in non-bi-orderable 3-manifold groups.

We prove the hyperbolization theorem for punctured torus bundles and two-bridge link complements by decomposing them into ideal tetrahedra which are then given hyperbolic structures, following Rivin's volume maximization principle.

Study of quaternion Azumaya algebras in hyperbolic once-punctured torus bundles.

New definition of twisted 1-loop invariant using Ptolemy coordinates.

We describe a class $\mathcal{C}$ of punctured torus bundles such that, for each $M \in \mathcal{C}$, all but finitely many Dehn fillings on $M$ are virtually Haken. We show that $\mathcal{C}$ contains infinitely many commensurability classes, and we give evidence that $\mathcal{C}$ includes representatives of ``most''…

Study character varieties of hyperbolic 3-manifolds using bundle methods.

The paper supports a conjecture about a vanishing identity for certain 3-manifolds.

We determine the PSL_2(C) and SL_2(C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them …

If M is a hyperbolic once-punctured torus bundle over the circle, then the trace field of M has no real places.

Let M be a hyperbolic manifold of finite volume which fibers over the circle with fiber a once punctured torus, and let S be an arbitrary incompressible surface in M. We determine the characteristic JSJ-subpair of M-S and show, in particular, that the guts of (M,S) is empty.

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…

In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defin…

Minimal ideal triangulations studied for hyperbolic 3-manifolds.

We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.

To each once-punctured-torus bundle, $T_φ$, over the circle with pseudo-Anosov monodromy $φ$, there are associated two tessellations of the complex plane: one, $Δ(φ)$, is (the projection from $\infty$ of) the triangulation of a horosphere at $\infty$ induced by the canonical decomposition into ideal tetrahedra, and the…

New isolated geometric triangulations found in once-punctured torus bundles.

Supose that $Y$ is a lens space with $|H_1(Y; \mathbb{Z})|$ prime, and $Y$ does not contain a genus one fibered knot. We show that $Y$ contains a knot whose exterior is a once-punctured torus bundle if and only if $Y$ is the result of $p/q$-surgery on the trefoil. This partially answers a question posed by Ken Baker in…

We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.

New examples of surface bundles found over surfaces.

Let $M$ be a once-punctured torus bundle over $S^1$ with monodromy $h$. We show that, under certain hypotheses on $h$, "most" Dehn-fillings of $M$ (in some cases all but finitely many) are virtually $\mathbb{Z}$-representable. We apply our results to show that surgeries on the figure-eight knot with even numerator are …

We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.

Our results complement D. Calegari's result that there are no hyperbolic once-punctured torus bundles over $S^1$ with trace field having real place. We exhibit several infinite families of pairs $(-χ, p)$ such that there exist hyperbolic surface bundles with over $S^1$ with fiber having $p$ punctures and Euler characte…

We determine the genus one fibered knots in lens spaces that have tunnel number one. We also show that every tunnel number one, once-punctured torus bundle is the result of Dehn filling a component of the Whitehead link in the 3-sphere.

We determine the non-null homologous knots in lens spaces whose exteriors contain properly embedded once-punctured tori. All such knots arise as surgeries on the Whitehead link and are grid number 1 in their lens spaces. As a corollary, we classify once-punctured torus bundles that admit a lens space filling.

Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two…

The study counts curves on a once-punctured torus with self-intersections.

Extends Dynnikov coordinates to punctured torus.

In this paper we give a necessary and sufficient condition in which a sequence of Kleinian punctured torus groups converges. This result tells us that every exotically convergent sequence of Kleinian punctured torus groups is obtained by the method due to Anderson and Canary. Thus we obtain a complete description of th…

Thurston's ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free t…

Researchers compute quantum invariant for four-puncture sphere, verifying volume conjecture.

We show that the space of Kleinian punctured torus groups is not locally connected.

Study shows the volume of convex core for once-punctured torus groups is close to a fixed value.

We find the exact values of complexity for an infinite series of 3-manifolds. Namely, by calculating hyperbolic volumes, we show that c(N_n)=2n, where $c$ is the complexity of a 3-manifold and N_n is the total space of the punctured torus bundle over S^1 with monodromy 2&1 1&1 ^n$. We also apply a recent result of Matv…

Connected graph for twice-punctured torus curves.

If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $π_1N$ and $π_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2\rtimes\mathbb{Z}$ are distinguished from one another by their profinite…

This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to expl…

Study quantized SL2-character variety of a once-punctured torus, finding three Coulomb branch isomorphisms.

A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial compu…

New algebra for twice-punctured torus curves.

We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an Epstein-Penner convex hull constructio…

Study Agol cycles on 2-punctured torus and 5-punctured sphere, finding new dilatation formula.

Paper finds infinite family of minimal triangulations for complex 3D shapes.

Study earthquake deformations on a once-punctured torus.

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.