arXiv research

Covering maps help determine ideal embeddings in homogeneous spaces.

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 …

Minimal ideal triangulations studied for hyperbolic 3-manifolds.

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

We define enhancements of the quandle counting invariant for knots and links with a finite labeling quandle Q embedded in the quandle of units of a Lie algebra \mathfrak{a} using Lie ideals. We provide examples demonstrating that the enhancement is stronger than the associated unenhanced counting invariant.

Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation be…

New theorem shows embedding impossibility for certain 3-manifolds.

We give an explicit basis $\mathcal{B}$ of the quotient of the Kauffman bracket skein algebra $\mathcal{S} (Σ)$ on a surface $Σ$ by the square of an augmentation ideal. As an application, it induces two kinds of finite type invariants of links in a handle body in the sense of Le. Moreover, we construct an embedding of …

Characterizes infinite ideal polyhedra in hyperbolic 3-space and proves their existence and rigidity.

A submanifold in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Moebius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m>3 which are Moebius homogeneous. There are three cla…

We prove the Goldman-Parker Conjecture: A complex hyperbolic ideal triangle group is directly embedded in PU(2,1) if and only if the product of its three standard generators is not elliptic. We also prove that such a group is indiscrete if the product of its three standard generators is elliptic. A novel feature of thi…

Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define …

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular pol…

A fast graph embedding method for large graphs.

First I will explain my motivation to introduce the $δ$-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of $δ$-invariants to several areas in mathematics. Finally, I will present two optimal inequalities …

We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…

Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2π$. The first examples appearing in this context are vertical geodesic planes and Scherk…

Study finds knots with ideal length need not have smallest volume.

Study characterizes Uniswap v3 liquidity pools using transaction graphs and identifies ideal trading conditions.

We present explicit geometric decompositions of the complement of tiling links, which are alternating links whose projection graphs are uniform tilings of the 2-sphere, the Euclidean plane or the hyperbolic plane. This requires generalizing the angle structures program of Casson and Rivin for triangulations with a mixt…

Study metric learning from limited preference comparisons, showing how low-dimensional structure can still reveal metric information.

The delta invariant of curves on rational surfaces is calculated using embedded topological and analytic methods.

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…

A new query embedding method improves KB performance on complex queries.

EPNE models evolving network patterns for better predictions.

Transformers learn to recall with non-orthogonal embeddings in realistic settings.

Study how knots occupy space using topological methods.

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 fr…

Kauffman bracket skein algebra structure on surfaces defined.

Paper proposes GrokTransfer to eliminate delayed generalization in neural networks.

Study ideals in the Goldman Lie algebra $S$ by mapping to a simpler algebra.

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

We give a simple method to find ideal points of the character variety of a 3-manifold from an ideal triangulation.

Paper provides new Alexander ideal-based obstruction to 0-concordance of knotted surfaces.

Short proof for ideal polygons with near optimal orthogeodesic decomposition.

A method uses CG to create efficient channels for ideal observers.

The paper explores genericity of triples of antipodal chambers in affine buildings.

Study introduces dynamical ideals for non-commutative rings and classifies knots and links.

We investigate the rigidity of hyperbolic cone metrics on $3$-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…

Defines timelike ideal boundary for non-positively curved Lorentzian spaces.

CNEs improve network embeddings by adding structural information.

A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behavio…

Defines ideal simplicial volume for manifolds with boundary, showing it's bounded by classical volume.

The paper studies dynamical properties in semigroups modulo ideals.

The notion of ideal immersions was introduced by the author in 1990s. Roughly speaking, an ideal immersion of a Riemannian manifold into a real space form is a nice isometric immersion which produces the least possible amount of tension from the ambient space at each point. In this paper, we classify all ideal hypersur…

New formula calculates volumes of ideal hyperbolic drums.

Study of combinatorial Calabi flow on ideal circle patterns.

The paper studies deformations of Lie ideals in Lie algebras.