arXiv research

Geometric DEC solves Poisson on general triangulations.

New triangulations encode flows with vanishing polynomial.

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1…

Quasitoric manifolds, introduced by M. Davis and T. Januskiewicz in 1991, are topological generalizations of smooth complex projective spaces. In 1992, Banchoff and Kühnel constructed a 10-vertex equilibrium triangulations of $\CP^2$. We generalize this construction for quasitoric manifolds and construct some equilibri…

Minimal Delaunay triangulations on hyperbolic surfaces have linear number of vertices.

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 ne…

Every noncompact surface has a 3-rigid triangulation.

The paper generates triangulations of 2-knot complements via spinning 1-knots.

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-…

Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic met…

The paper solves pentagon equations using triangulations and edge transformations.

It is important to have fast and 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 …

This note popularizes a proof for 3-manifold triangulations and spines.

Method samples triangulations of manifolds using biased random walks.

Extends circle pattern theorem to quasi-simplicial triangulations.

Proved contractibility of geodesic triangulation space on hyperbolic surfaces.

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…

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

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be …

The paper studies triangulations of surfaces up to branched transit equivalences, proving equivalence conditions and foliations.

Let $X$ be a real analytic orbifold. Then each stratum of $X$ is a subanalytic subset of $X$. We show that $X$ has a unique subanalytic triangulation compatible with the strata of $X$. We also show that every ${\rm C}^r$-orbifold, $1\leq r\leq \infty$, has a real analytic structure. This allows us to triangulate differ…

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

The paper triangulates Heisenberg groups with horizontal and straight simplexes.

Connected flip graphs for triangulations on hyperbolic surfaces.

PointTriNet generates 3D triangulations from point clouds efficiently and scalably.

Let $C(L)$ be the right-angled Coxeter group defined by an abstract triangulation $L$ of $\mathbb{S}^2$. We show that $C(L)$ is isomorphic to a hyperbolic right-angled reflection group if and only if $L$ can be realized as an acute triangulation. The proof relies on the theory of CAT(-1) spaces. A corollary is that an …

Essential triangulations connect via specific moves in 3-manifolds.

Random veering triangulations are not geometric in most cases.

Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…

New triangulations show harder skeletons for hyperbolic orbifolds.

New findings on strong convexity in triangulations of convex polygons.

A polynomial invariant for veering triangulations helps in understanding 3-manifold fibers.

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

This paper uses the technology of weighted and regular triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Regular triangulations are studied in some detail, including flip algorithms. The Laplacian is then studied as an operator on functions of the vertices as a generalized wei…

Efficient triangulations help in understanding 3-manifold boundaries.

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 fundament…

A 6-regular triangulation for hyperbolic plane created.

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

Thurston's jiggling lemma simplifies triangulations.

Geometric triangulations can be transformed by bistellar moves.

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…

We present the mathematical background of a software package that computes triangulations of mapping tori of surface homeomorphisms, suitable for Jeff Weeks's program SnapPea. It consists of two programs. jmt computes triangulations and prints them in a human-readable format. jsnap converts this format into SnapPea's t…

New matrices link point motions to braid groups.

Machine learning identifies 3-manifold triangulations using isomorphism signatures.

We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …

Minimal triangulations for 229 hyperbolic census knots discovered.

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…

Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, givin…