Minimal triangulations of circle bundles, circular permutations and binary Chern cocycle

Minimal triangulations of sphere bundles over the circle

For integers $d \geq 2$ and $ε= 0$ or 1, let $S^{1, d - 1}(ε)$ denote the sphere product $S^{1} \times S^{d - 1}$ if $ε= 0$ and the twisted $S^{d - 1}$ bundle over $S^{1}$ if $ε= 1$. The main results of this paper are: (a) if $d \equiv ε$ (mod 2) then $S^{1, d - 1}(ε)$ has a unique minimal triangulation using $2d + 3$ vertices, and (b) if $d \equiv 1 - ε$ (mod 2) then $S^{1, d - 1}(ε)$ has minimal triangulations (not unique) using $2d + 4$ vertices. The second result confirms a recent conjecture of Lutz. The first result provides the first known infinite family of closed manifolds (other than …

Veering triangulations and Cannon-Thurston maps

Any hyperbolic surface bundle over the circle gives rise to a continuous surjection from the circle to the sphere, by work of Cannon and Thurston. We prove that the order in which this surjection fills out the sphere is dictated by a natural triangulation of the surface bundle (introduced by Agol) when all singularities of the invariant foliations are at punctures of the fiber.

Circle packings on surfaces with projective structures

The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ_g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or -1 on the surface depending on whether g=0, 1 or \ge 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface Σ_g where ci…

带锥奇点的曲面上的无限圆填充

Infinite circle packings on surfaces with conical singularities

本文证明了在满足 Gauss-Bonnet 不等式的带穿孔曲面上，给定无限三角剖分和小于 π 的目标锥角，必存在支持相应圆填充的双曲度量；并构造出同一共形类中具有无穷多个锥双曲结构的反例，与闭曲面情形的 Kojima-Mizushima-Tan 猜想形成鲜明对比。

圆堆积的代数方法

An algebraic approach to circle packing

本文针对曲面的特定三角剖分，提出用多项式方程组求解圆堆积的代数方法，还建立了旋量方法与经典欧氏几何的联系。

The average dual surface of a cohomology class and minimal simplicial decompositions of infinitely many lens spaces

Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component. They are also natural Poincaré duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed cohomology class in a simplicial poset the average Euler characteristic of the associated discrete normal surfaces only depends on the $f$-vector of the triangulation. As an application we determine the minimum simplicial poset representations, also known as crystallizations, of lens spaces $L(2k,q),$ where $2k=qr+1.$ Higher dimensional analogs of discrete normal surfaces …

Ideal triangulations of 3-manifolds II: taut and angle structures

This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program SNAPPEA of Weeks and the program SNAP of Coulson, Goodman, Hodgson and Neumann. Casson has also written a program to find hyperbolic structures on such 3-manifolds, by solving Thurston's hyperbolic gluing equations for ideal triangulations. In this second paper, we study the ques…

Counting triangles formula for the first Chern class of a circle bundle

We consider the problem of the combinatorial computation of the first Chern class of a circle bundle. N.Mnev found such a formula in terms of canonical shellings. It represents certain invariant of a triangulation computed by analyzing cyclic word in 3-character alphabet associated to the bundle. This curvature is a kind of discretization of Konstevich's curvature differential 2-form. We find a new expression of Mnev's curvature by counting triangles in a cyclic word. Our formula is different from that of Mnev. In particular, it is cyclically invariant by its very form. We present also some sa…

Circle packings on surfaces with projective structures and uniformization

Let Σ_g be a closed orientable surface of genus g \geq 2 and τa graph on Σ_g with one vertex which lifts to a triangulation of the universal cover. We have shown that the cross ratio parameter space \mathcal{C}_τassociated with τ, which can be identified with the set of all pairs of a projective structure and a circle packing on it with nerve isotopic to τ, is homeomorphic to \mathbb{R}^{6g-6}, and moreover that the forgetting map of \mathcal{C}_τto the space of projective structures is injective. In this paper, we show that the composition of the forgetting map with the uniformization from \m…

The Meyer function on the handlebody group

We give an explicit formula for the signature of handlebody bundles over the circle in terms of the homological monodromy. This gives a cobounding function of Meyer's signature cocycle on the mapping class group of a $3$-dimensional handlebody, i.e., the handlebody group. As an application, we give a topological interpretation for the generator of the first cohomology group of the hyperelliptic handlebody group.

Computing a Link Diagram from its Exterior

A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by planar diagrams, whereas their exteriors, which are compact 3-manifolds with torus boundary, are encoded by triangulations. Here, we give the first practical algorithm for finding a diagram of a knot given a triangulation of its exterior. Our method applies to links as well as knots, allows us to recover links with hundreds of crossings. We use it to find the fi…

Discrete surfaces with length and area and minimal fillings of the circle

We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian surfaces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrical…