Related papers: Perfect t-embeddings and Lozenge Tilings

A complete embedding is a symplectic embedding $\iota:Y\to M$ of a geometrically bounded symplectic manifold $Y$ into another geometrically bounded symplectic manifold $M$ of the same dimension. When $Y$ satisfies an additional finiteness…

We discuss some specializations of the frames of flat orthonormal frame bundles over geometries of indefinite signature, and the resulting symmetries of families of embedded Riemannian or pseudo-Riemannian geometries. The specializations…

We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical…

Denote by $T_k$ the generalised triangle, a $k$-uniform hypergraph on vertex set $\{1,2,\dots,2k-1\}$ with three edges $\{1,\dots,k-1,k\}$,$\{1,\dots,k-1,k+1\}$ and $\{k,k+1,\dots,2k-1\}$. Recently, Bowtell, Kathapurkar, Morrison and…

Totally symmetric self-complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well-known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in one-twelfth of…

"Flat origami" refers to the folding of flat, zero-curvature paper such that the finished object lies in a plane. Mathematically, flat origami consists of a continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ along…

Lattices and their underlying symmetries play a central role in determining the physical properties and applications of many natural and engineered materials. By bridging the lattice geometry and rigid-folding kinematics, this study…

Let $G=(V,E)$ be a matching-covered graph, denote by $P$ its perfect matching polytope, and by $L$ the integer lattice generated by the integral points in $P$. In this paper, we give short, polyhedral proofs for two difficult results…

We provide two constructions for $t$ edge-disjoint maximal outerplanar graphs on every number of $n \geq 4t$ vertices. The bound on the minimum number of vertices is tight. These constructions yield the existence of optimal…

We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our…

This article is the continuation of the first named author work "On maximal totally real embeddings". For real analytic compact manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent…

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and,…

A foundational result in origami mathematics is Kawasaki and Justin's simple, efficient characterization of flat foldability for unassigned single-vertex crease patterns (where each crease can fold mountain or valley) on flat material. This…

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the…

We introduce Gaussian quadrature rules for spline spaces that are frequently used in Galerkin discretizations to build mass and stiffness matrices. By definition, these spaces are of even degrees. The optimal quadrature rules we recently…

We show on non-flat but critical s-embeddings the celebrated convergence of the interface curves of the critical FK Ising model to an $\operatorname{SLE}_{16/3}$ curve, using discrete complex analytic techniques first used in…

We survey and enlarge the known mappings of the 16-vertex model, with emphasis on mappings between the even and odd 8-vertex subcases of the general model, also giving new mappings between these models, valid on finite toroidal lattices. In…

It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more…

We consider the problem of immersing the complete digraph on t vertices in a simple digraph. For Eulerian digraphs, we show that such an immersion always exists whenever minimum degree is at least t(t-1), and for t at most 4 minimum degree…

Flat origami studies straight line, planar graphs $C=(V,E)$ drawn on a region $R\subset\mathbb{R}^2$ that can act as crease patterns to map, or fold, $R$ into $\mathbb{R}^2$ in a way that is continuous and a piecewise isometry exactly on…