A method to find ideal points from ideal triangulations

Ideal triangulations of finite volume hyperbolic 3-manifolds

This paper was withdrawn (temporarily?) by the author since an error needs to be corrected.

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…

A new family of minimal ideal triangulations of cusped hyperbolic 3-manifolds

Previous work of the authors with Bus Jaco determined a lower bound on the complexity of cusped hyperbolic 3-manifolds and showed that it is attained by the monodromy ideal triangulations of once-punctured torus bundles. This paper exhibits an infinite family of minimal ideal triangulations of Dehn fillings on the link $8^3_9$ that also attain this lower bound on complexity.

通过有序理想三角剖分对闭 $3$-流形的组合描述

Combinatorial description of closed $3$-manifolds via ordered ideal triangulations

本文通过有序理想三角剖分及有序Pachner $2$-$3$和$0$-$2$移动，给出了闭$3$-流形的组合描述。

Geodesic ideal triangulations exist virtually

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups.

A calculus for ideal triangulations of three-manifolds with embedded arcs

Refining the notion of an ideal triangulation of a compact three-manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,a), where M is a three-manifold and a is a collection of properly embedded arcs. We also show that certain well-understood combinatorial moves are sufficient to relate to each other any two refined triangulations representing the same (M,a). Our proof does not assume the Matveev-Pergallini calculus for ideal triangulations, and actually easily implies this calculus.

Pseudo-Developing Maps for Ideal Triangulations I: Essential Edges and Generalised Hyperbolic Gluing Equations

Let N be a topologically finite, orientable 3-manifold with ideal triangulation. We show that if there is a solution to the hyperbolic gluing equations, then all edges in the triangulation are essential. This result is extended to a generalisation of the hyperbolic gluing equations, which enables the construction of hyperbolic cone-manifold structures on N with singular locus contained in the 1-skeleton of the triangulation.

Essential loops in taut ideal triangulations

In this note we combinatorialise a technique of Novikov. We use this to prove that, in a three-manifold equipped with a taut ideal triangulation, any vertical or normal loop is essential in the fundamental group.

Ideal Triangulations of Pseudo-Anosov Mapping Tori

We show how to construct an ideal triangulation of a mapping torus of a pseudo-Anosov map punctured along the singular fibers. This gives rise to a new conjugacy invariant of mapping classes, and a new proof of a theorem of Farb-Leininger-Margalit. The approach in this paper is based on ideas of Hamenstadt.

Negatively Oriented Ideal Triangulations and a Proof of Thurston's Hyperbolic Dehn Filling Theorem

We give a complete proof of Thurston's celebrated hyperbolic Dehn filling theorem, following the ideal triangulation approach of Thurston and Neumann-Zagier. We avoid to assume that a genuine ideal triangulation always exists, using only a partially flat one, obtained by subdividing an Epstein-Penner decomposition. This forces us to deal with negatively oriented tetrahedra. Our analysis of the set of hyperbolic Dehn filling coefficients is elementary and self-contained. In particular, it does not assume smoothness of the complete point in the variety of deformations.

The 3D index of an ideal triangulation and angle structures

The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with $r$ torii boundary components. For a fixed $2r$ tuple of integers, the index takes values in the set of $q$-series with integer coefficients. Our goal is to give an axiomatic definition of the tetrahedron index, and a proof that the domain of the 3D index consists precisely of the set of ideal triangulations that support an index structure. The latter is a generalization of a strict angle structure. We also prove that the 3D index is invariant under 3-2 moves, but not in ge…

Constructing Infinitely Many Geometric Triangulations Of The Figure Eight Knot Complement

This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold.

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T, is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.

Efficient triangulations and boundary slopes

For a compact, irreducible, $\partial$-irreducible, an-annular bounded 3-manifold $M\ne\mathbb{B}^3$, then any triangulation $\mathcal{T}$ of $M$ can be modified to an ideal triangulation $\mathcal{T}^*$ of $\stackrel{\circ}{M}$. We use the inverse relationship of crushing a triangulation along a normal surface and that of inflating an ideal triangulation to introduce and study boundary-efficient triangulations and end-efficient ideal triangulations. We prove that the topological conditions necessary for a compact 3-manifold $M$ admitting an annular-efficient triangulation are sufficient to mo…

理想三角剖分与单穿孔曲面丛

Ideal Triangulations and Once-Punctured Surface Bundles

本文证明了在具有单个环面边界分量的无环面无柱面不可约紧致3-流形中，存在使纤维旋转正规化的理想三角剖分，并给出了构造该三角剖分的算法。

On the ideal triangulation graph of a punctured surface

We study the ideal triangulation graph $T(S)$ of a punctured surface $S$ of finite type. We show that if $S$ is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of $S$ into the simplicial automorphism group of $T(S)$ is an isomorphism. We also show that under the same conditions on $S$, the graph $T(S)$ equipped with its natural simplicial metric is not Gromov hyperbolic. Thus, from the point of view of Gromov hyperbolicity, the situation of $T(S)$ is different from that of the curve complex of $S$.

Tropical varieties associated to ideal triangulations: The Whitehead link complement

This paper illustrates a computational approach to Culler-Morgan-Shalen theory using ideal triangulations, spun-normal surfaces and tropical geometry. Certain affine algebraic sets associated to the Whitehead link complement as well as their logarithmic limit sets are computed. The projective solution space of spun-normal surface theory is related to the space of incompressible surfaces and to the unit ball of the Thurston norm. It is shown that all boundary curves of the Whitehead link complement are strongly detected by its character variety. The specific results obtained can be used to stud…

0-efficient triangulations of 3-manifolds

0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S^3, RP^3 or L(3,1). Similarly, any triangulation of a compact, orientable, irreducible, boundary-irreducible 3-manifold can be modified to a 0-efficient triangulation. The notion of a 0-efficient ideal triangulation is defined. It is shown if M is a compact, orientable, irreducible, boundary-irreducible 3-manifold having no essential annuli and distinct…

On ideal triangulations of surfaces up to branched transit equivalences

We consider triangulations of closed surfaces S with a given set of vertices V; every triangulation can be branched that is enhanced to a Delta-complex. Branched triangulations are considered up to the b-transit equivalence generated by b-flips (i.e. branched diagonal exchanges) and isotopy keeping V point-wise fixed. We extend a well known connectivity result for `naked' triangulations; in particular in the generic case when S has negative Euler-Poincare' characteristic c(S), we show that branched triangulations are equivalent to each other if c(S) is even, while this holds also for odd c(S) …

Ideal triangulations of 3-manifolds up to decorated transit equivalences

We consider 3-dimensional pseudo-manifolds M with a given set of marked point V such that M-V is the interior of a compact 3-manifold with boundary. An ideal triangulation T of (M, V ) has V as its set of vertices. A branching (T, b) enhances T to a Delta-complex. Branched triangulations of (M, V ) are considered up to the b-transit equivalence generated by isotopy and ideal branched moves which keep V pointwise fixed. We extend a well known connectivity result for naked triangulations by showing that branched ideal triangulations of (M, V) are equivalent to each other. A pre-branching is a sy…

论孤立几何三角剖分

On Isolated Geometric Triangulations

本文定义孤立几何三角剖分，构造无限一次穿孔环面丛族证明其非小流形独有，回应了几何三角剖分子图的连通性相关问题。

Triangulations of hyperbolic 3-manifolds admitting strict angle structures

It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every knot or link complement in the 3-sphere has such a triangulation. We also give an example of a triangulation without a strict angle structure, where the obstruction is related to the homology hypothesis…

Generalized Mom-structures and ideal triangulations of 3-manifolds with non-spherical boundary

The so-called Mom-structures on hyperbolic cusped 3-manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of 3-manifolds with non-empty boundary that does not have spherical components. We then describe a certain relation between such generalized Mom-structures, called protoMom-structures, internal on a fixed 3-manifold N, and ideal triangulations of N; in addition, in the case of non-closed hyperbolic manifolds without annular cus…

无限黎曼面的有界理想三角剖分

Bounded ideal triangulations of infinite Riemann surfaces

本文提出无限黎曼面的有界理想三角剖分概念，构造了相应Teichmüller空间的实解析参数化，并证明有界几何且可数穿孔的曲面属于此类，弥补了Fenchel-Nielsen参数化在实解析性上的不足。

Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots

We construct a new infinite family of ideal triangulations and H-triangulations for the complements of twist knots, using a method originating from Thurston. These triangulations provide a new upper bound for the Matveev complexity of twist knot complements. We then prove that these ideal triangulations are geometric. The proof uses techniques of Futer and the second author, which consist in studying the volume functional on the polyhedron of angle structures. Finally, we use these triangulations to compute explicitly the partition function of the Teichmüller TQFT and to prove the associated v…

Delta-groupoids and ideal triangulations

A Delta-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. By considering two simplest examples coming from knot theory, we illustrate how can one associate a Delta-groupoid to an ideal triangulation of a three-manifold. We also describe in detail the rings associated with the Delta-groupoids of these examples.

Ideal triangulations and geometric transitions

Thurston introduced a technique for finding and deforming three-dimensional hyperbolic structures by gluing together ideal tetrahedra. We generalize this technique to study families of geometric structures that transition from hyperbolic to anti de Sitter (AdS) geometry. Our approach involves solving Thurston's gluing equations over several different shape parameter algebras. In the case of a punctured torus bundle with Anosov monodromy, we identify two components of real solutions for which there are always nearby positively oriented solutions over both the complex and pseudo-complex numbers.…

Ideal triangulations of 3--manifolds I: spun normal surface theory

In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map from spun normal surface theory to the homology classes of boundary loops of the 3-manifold and prove the boundary map has image of finite index. Spun normal surfaces give a natural way of representing properly embedded and immersed essential surfaces in a 3-manifold with tori and …

超曲面螺旋三角化中的剪切边界

Bounding shears of spiralling triangulations on hyperbolic surfaces

本研究证明双曲面可通过具有有界剪切参数的理想三角化，其上界与曲面拓扑的对数相关。

Ideal Triangulation and Disk Unfolding of a Singular Flat Surface

An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface. Using the fact that each pair of points in a surface has a finite number of geodesics having length $\leq L$ connecting them, where $L$ is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point. Also, we prove that such a surface contains a finite number of geodesics wh…

没有正旋转三角剖分的双曲流形

Hyperbolic manifolds without positive spun triangulations

研究由闭双曲三维流形和简单闭测地线组成的对，利用Choi结果给出无前述正旋转理想三角剖分的例子，如Vol3及相关测地线，为Vol3无此类三角剖分猜想提供证据。

1-efficient triangulations and the index of a cusped hyperbolic 3-manifold

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index structure if and only if T is 1-efficient and (b) if M is hyperbolic, it has a canonical set of 1-efficient ideal triangulations related by 2-3 and 0-2 moves which preserve the 3D index. We illustrate our results with several examples.

连通本质三角剖分 II：仅通过2-3变换

Connecting essential triangulations II: via 2-3 moves only

本研究针对万有覆叠有无穷多边界分支的流形M，证明其本质理想三角剖分集在忽略无保本质性2-3变换的剖分时可仅通过2-3、3-2变换连通，加入V-变换及其逆则全体连通，且结论对L-本质三角剖分也成立。

Random ideal triangulations and the Weil-Petersson distance between finite degree covers of punctured Riemann surfaces

We prove that any two finite-area non-compact hyperbolic Riemann surfaces S and T have finite covers that are arbitrarily close in the normalized Weil-Petersson metric, where we normalize by dividing the square of the metric by the area of the surface. In the case where T is the modular surface this reduces to showing that S has a finite cover with a proper ideal triangulation where most of the shear coordinates are small; we will construct such a cover out of a random collection of immersed ideal triangles in S.

Bounds on Pachner moves and systoles of cusped 3-manifolds

Any two geometric ideal triangulations of a cusped complete hyperbolic $3$-manifold $M$ are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check if two hyperbolic knots are equivalent, given geometric ideal triangulations of their complements. Given a geometric ideal triangulation of $M$, we also give a lower bound on the systole length of $M$ in terms of the number of tetrahedra and a lower …

The ideal Thurston-Andreev theorem and triangulation production

This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be the gradient of an objective function related in a rather magical way to hyperbolic volume.

Quantum invariants of closed framed $3$-manifolds based on ideal triangulations

We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of $3$-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner $(2,3)$ move of ideal triangulations and the non-involutive…

Geodesic triangulations exist for cusped Platonic manifolds

We show that if a cusped hyperbolic manifold is Platonic, i.e., can be decomposed into isometric Platonic solids, it can also be decomposed into geodesic ideal tetrahedra.

几何超理想三角剖分的局部组合判据：基于组合Ricci流

Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow

本文通过组合Ricci流为双曲截断四面体实现理想三角剖分提供局部组合判据，涵盖度为7和6的边条件，证明零曲率超理想度量的唯一性与指数收敛。

An application of non-positively curved cubings of alternating links

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that the hyperbolicity equations for those triangulations for hyperbolic alternating links have solutions corresponding to the complete hyperbolic structures. Since the ideal triangulations considered in this paper are often used in the study of the volume conjecture, this result has a …

Connectivity of triangulations without degree one edges under 2-3 and 3-2 moves

Matveev and Piergallini independently showed that, with a small number of known exceptions, any triangulation of a three-manifold can be transformed into any other triangulation of the same three-manifold with the same number of vertices, via a sequence of 2-3 and 3-2 moves. We can interpret this as showing that the Pachner graph of such triangulations is connected. In this paper, we extend this result to show that (again with a small number of known exceptions), the subgraph of the Pachner graph consisting of triangulations without degree one edges is also connected, for single-vertex triangu…

几何三角剖分与高扭曲链环

Geometric triangulations and highly twisted links

针对带尖点双曲3-流形的几何三角剖分猜想，证明充分高扭曲纽结存在几何三角剖分，并拓展已有工作给出无穷族实例的量化结果。

“魔幻流形”的三角剖分与普查纽结族

Triangulations of the `magic manifold' and families of census knots

本文通过构造魔幻流形的五种理想三角剖分，为所有部分填充得到低复杂度三角剖分，恢复229个普查纽结的最小剖分，并将其分为42个可无限扩展的扭转族。

Delta-groupoids in knot theory

A Delta-groupoid is an algebraic structure which axiomitizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of Delta-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, one can associate a Delta-groupoid to ideal triangulations of knot complements. It is also possible to define a homology theory of Delta-groupoids. The constructions are illustrated by examples coming from knot theory.

Non-peripheral ideal decompositions of alternating knots

An ideal triangulation $\mathcal{T}$ of a hyperbolic 3-manifold $M$ with one cusp is non-peripheral if no edge of $\mathcal{T}$ is homotopic to a curve in the boundary torus of $M$. For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of $M$. A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are ideal triangulations that use 4 (resp., 5) ideal tetrahedra per crossing. Our main result is that these ideal triangulations are non-peripheral for all planar, reduced, alternat…

3-流形三角剖分的辛基

A symplectic basis for 3-manifold triangulations

本文为Neumann-Zagier辛向量空间提供了基于Heegaard曲面曲线相交的几何解释，并给出了构造辛基的算法。

Canonical triangulations of Dehn fillings

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold. As an example, we treat all hyperbolic fillings on one cusp of the Whitehead link complement.

The universal quantum invariant and colored ideal triangulations

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has…

Infinitely many virtual geometric triangulations

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 …

Ideally, all infinite type surfaces can be triangulated

We show that any surface of infinite type admits an ideal triangulation. Furthermore, we show that a set of disjoint arcs can be completed into a triangulation if and only if, as a set, they intersect every simple closed curve a finite number of times.

k-Normal surfaces

Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: 1) a minimal triangulation of a closed irreducible or a bounded hyperbolic 3-manifold contains no non-trivial k-normal sphere; 2) every triangulation of a closed manifold with at least 2 tetrahedra contains some non-trivial normal surface; 3) every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, tria…

Triangulations into Groups

If a (cusped) surface S admits an ideal triangulation T with no shears, we show an efficient algorithm to give S as a quotient of hypebolic plane by a subgroup of PSL(2, Z). The algorithm runs in time O(n log n), where n is the number of triangles in the triangulation T. The algorithm generalizes to producing fundamental groups of general surfaces and geometric manifolds of higher dimension.

Spin structures on 3-manifolds via arbitrary triangulations

Let M be an oriented compact 3-manifold and let T be a (loose) triangulation of M, with ideal vertices at the components of the boundary of M and possibly internal vertices. We show that any spin structure s on M can be encoded by extra combinatorial structures on T. We then analyze how to change these extra structures on T, and T itself, without changing s, thereby getting a combinatorial realization, in the usual "objects/moves" sense, of the set of all pairs (M,s). Our moves have a local nature, except one, that has a global flavour but is explicitly described anyway. We also provide an alt…

结三角剖分的积分面-边对偶性

Integral face-edge duality for knot triangulations

本文通过上链构造建立结三角剖分中面矩阵与粘合矩阵的对偶关系，证明零化度加倍及FAMED恒等式，并给出Wong猜想的拟议证明。