arXiv research

Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.

Combinatorial description of 3-manifolds using ordered triangulations.

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, …

We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian manifolds) to obtain triangulations with these properties under various hypotheses…

Moving between 3-manifold triangulations is NP-hard

3-manifold triangulation can be reconstructed from its intersection matrix.

The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and P^2-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with part…

In this paper, we describe geometrical constructions to obtain triangulations of connected sums of closed orientable triangulated 3-manifolds. Using these constructions, we show that it takes time polynomial in the number of tetrahedra to check if a closed orientable 3-manifold, equipped with a minimal triangulation, i…

Proof of existence for ideal triangulations that normalize fibers in certain 3-manifolds.

Geometrically interprets symplectic structure in 3-manifold triangulations.

Efficient triangulations help in understanding 3-manifold boundaries.

Paper proves Luo's conjecture for 3D triangulated manifolds.

In this thesis, we use normal surface theory to understand certain properties of minimal triangulations of compact orientable 3-manifolds. We describe the collapsing process of normal 2-spheres and disks. Using some geometrical constructions to take connected sums of triangulated 3-manifolds, we obtain the following re…

A triangulation of a compact 3-manifold is annular-efficient if it is 0-efficient and the only normal, incompressible annuli are thin edge-linking. If a compact 3-manifold has an annular-efficient triangulation, then it is irreducible, boundary-irreducible, and an-annular. Conversely, it is shown that for a compact, ir…

3-manifold triangulations are Golod and tight, proven through a topological characterization.

The paper explores algorithms to transform 3-manifold triangulations while controlling sparsity.

The paper finds canonical triangulations for specific 3-manifolds.

New method finds large counterexamples by selectively exploring triangulations.

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

In this paper, we explore minimal contact triangulations on contact 3-manifolds. We give many explicit examples of contact triangulations that are close to minimal ones. The main results of this article say that on any closed oriented 3-manifold the number of vertices for minimal contact triangulations for overtwisted …

The triangulation complexity of a closed orientable 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. The main theorem of the paper gives upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the…

Algorithm constructs triangulations for Heegaard splittings and related 3-manifolds.

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-…

Machine learning identifies 3-manifold triangulations using isomorphism signatures.

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 …

A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…

3-manifolds have covers with infinitely many ideal triangulations.

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting ta…

It is not completely unreasonable to expect that a computable function bounding the number of Pachner moves needed to change any triangulation of a given 3-manifold into any other triangulation of the same 3-manifold exists. In this paper we describe a procedure yielding an explicit formula for such a function if the 3…

Unimodal sequences of moves connect 3-manifold triangulations.

Highly twisted knots can be geometrically triangulated.

New method for computing hyperbolic structures on 3-manifolds with torus boundaries.

This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.

The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the co…

Finitely many pseudo-Anosov flows without perfect fits in a 3-manifold.

Previous work of the authors studies minimal triangulations of closed 3-manifolds using a characterisation of low degree edges, embedded layered solid torus subcomplexes and 1-dimensional $\mathbb{Z}_2$-cohomology. The underlying blueprint is now used in the study of minimal ideal triangulations. As an application, it …

This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L(4k, 2k-1)$ and the generalised quaternionic space $S^3/Q_{4k}$ have complexity $k,…

New quantum invariant for framed 3-manifolds using ideal triangulations.

Proves unique hyperbolic metric for 3-manifolds with ideal triangulation.

In graph theory, as well as in 3-manifold topology, there exist several width-type parameters to describe how "simple" or "thin" a given graph or 3-manifold is. These parameters, such as pathwidth or treewidth for graphs, or the concept of thin position for 3-manifolds, play an important role when studying algorithmic …

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

A celebrated result concerning triangulations of a given closed 3-manifold is that any two triangulations with the same number of vertices are connected by a sequence of so-called 2-3 and 3-2 moves. A similar result is known for ideal triangulations of topologically finite non-compact 3-manifolds. These results build o…

Researchers prove a conjecture linking 1-loop invariants to torsion for fibered 3-manifolds.

It is well known that a triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known …

It is important to have effective methods for simplifying 3-manifold triangulations without losing any topological information. In theory this is difficult: we might need to make a triangulation super-exponentially more complex before we can make it smaller than its original size. Here we present experimental work that…

It is not known whether there exists a computable function bounding the number of Pachner moves needed to connect any two triangulation of a compact 3-manifold. In this paper we find an explicit bound of this kind for all Haken 3-manifolds which contain no fibred submanifolds as strongly simple pieces of their JSJ-deco…

A universal branched 3-manifold $W$ characterizes Sol 3-manifolds.