arXiv research

The paper classifies and describes five-sided hyperbolic polyhedra with one ideal vertex.

Software finds ideal polyhedra with rational dihedral angles and volume maxima.

We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescrib…

New proof for global rigidity of vertex scaling on polyhedral surfaces.

Study laws of cosines and sines for hyperbolic shapes with ideal vertices.

H. Masur and J. Smillie proved precisely which singularity index lists arise from pseudo-Anosov mapping classes. In search of an analogous theorem for outer automorphisms of free groups, Handel and Mosher ask: Is each connected, simplicial, (2r-1)-vertex graph the ideal Whitehead graph of a fully irreducible outer auto…

We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of Handel-Mosher axis bundles that are not just a single axis, as well as of "nongeneric" behavior in the sense of the "trai…

Asymptotics of quantum $6j$ symbols corresponding to a hyperbolic tetrahedra is investigated and the first two leading terms are determined for the case that the tetrahedron has a ideal or ultra-ideal vertex. These terms are given by the volume and the determinant of the Gram matrix of the tetrahedron. A relation to th…

We give a simple sufficient condition for a spun-normal surface in an ideal triangulation to be incompressible, namely that it is a vertex surface with non-empty boundary which has a quadrilateral in each tetrahedron. While this condition is far from being necessary, it is powerful enough to give two new results: the e…

The paper explores how to find relevant vertices in one graph using another graph's attributes and structure.

A fast graph embedding method for large graphs.

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to…

The paper proves ideal triangulations and disk unfolding for singular flat surfaces.

The paper tackles VN with multiple vertices of interest and adversarial contamination.

We show how to construct, for each $r \geq 3$, an ageometric, fully irreducible $φ\in Out(F_r)$ whose ideal Whitehead graph is the complete graph on $2r-1$ vertices. This paper is the second in a series of three where we show that precisely eighteen of the twenty-one connected, simplicial, five-vertex graphs are ideal …

Study rigidity and volume optimization of hyperbolic polyhedra.

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 interpret a normal surface in a (singular) three-manifold in terms of the homology of a chain complex. This allows us to study the relation between normal surfaces and their quadrilateral co-ordinates. Specifically, we give a proof of an (unpublished) observation independently given by Casson and Rubinstein saying t…

The concept of a normal surface in a triangulated, compact 3-manifold was generalised by Thurston to a spun-normal surface in a non-compact 3-manifold with ideal triangulation. This paper defines a boundary curve map which takes a spun-normal surface to an element of the direct sum of the first homology groups of the v…

We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or …

New $3$-manifolds created from $4$-regular graphs with unique Eulerian cycles.

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 …

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 Pa…

A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of orthoschemes is known in hyperbolic geometry. Roughly speaking, complete orthosche…

The pinning ideal of multiloops is shown to be NP-complete.

Vertex distortion detects if a knot is unknot.

The study finds the bounds of vertex orbits in maps derived from specific lattices.

Given a vertex of interest in a network $G_1$, the vertex nomination problem seeks to find the corresponding vertex of interest (if it exists) in a second network $G_2$. A vertex nomination scheme produces a list of the vertices in $G_2$, ranked according to how likely they are judged to be the corresponding vertex of …

New maps on the plane with specific symmetry properties identified.

634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.

This paper shows semi-equivelar toroidal maps are vertex-transitive covers.

Defines formal vertex laws related to Lie conformal algebras.

New relations for vertex polynomial in graphs of any degree.

The study examines vertices in curves with singular points in the Euclidean plane.

Vertex distortion measures how far lattice knots deviate from straight lines.

For random graphs distributed according to stochastic blockmodels, a special case of latent position graphs, adjacency spectral embedding followed by appropriate vertex classification is asymptotically Bayes optimal; but this approach requires knowledge of and critically depends on the model dimension. In this paper, w…

We prove that there exists a geodesic trajectory on the dodecahedron from a vertex to itself that does not pass through any other vertex.

New model learns graph features for classification.

Research determines criteria for semi-regular tilings in hyperbolic space.

Proves a generalized Whitehead cut vertex lemma for tree groups.

Solves Skopenkov's problem on graph embedding criteria.

Efficient method for vertex embedding and community detection.

New invariants distinguish spatial graphs not previously possible.

Given a graph in which a few vertices are deemed interesting a priori, the vertex nomination task is to order the remaining vertices into a nomination list such that there is a concentration of interesting vertices at the top of the list. Previous work has yielded several approaches to this problem, with theoretical re…

The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a…

Proofs for Moon's theorem and its generalization.

Investigates the vertex curve of smooth surfaces in 3D space, connecting geometry and image analysis.

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We …