Related papers: Minimal triangulations for an infinite family of l…
We study lower bounds for the number of vertices in a PL-triangulation of a given manifold $M$. While most of the previous estimates are based on the dimension and the connectivity of $M$, we show that further information can be extracted…
The fundamental matrix and trifocal tensor are convenient algebraic representations of the epipolar geometry of two and three view configurations, respectively. The estimation of these entities is central to most reconstruction algorithms,…
We show that the minimum distance projection in the L1-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a…
A partial formula is provided to calculate the smallest number of vertices possible in a quadrangulation on the closed orientable 2-manifold of given genus. This extends the previously known partial formula due to N. Hartsfield and G.…
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…
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…
Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is…
Let T be a triangulation of S^3 containing a link L in its 1-skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces.
For a closed orientable connected 3-manifold $M$, its complexity $\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in its triangulations. Under the assumption that $M$ is prime (but not necessarily atoroidal), we…
In this paper we develop a method for studying tight contact structures on lens spaces. We then derive uniqueness and non-existence statements for tight contact structures with certain (half) Euler classes on lens spaces. We also prove that…
We define the notion of an approximate triangulation for a manifold $M$ embedded in euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in $M$ and use persistent homology to find a complex…
In 1983, Banchoff and Kuhnel constructed a minimal triangulation of $\CP^2$ with 9 vertices. $\CP^3$ was first triangulated by Bagchi and Datta in 2012 with 18 vertices. Known lower bound on number of vertices of a triangulation of $\CP^n$…
We introduce "$t$-LC triangulated manifolds" as those triangulations obtainable from a tree of $d$-simplices by recursively identifying two boundary $(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC notion…
We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting…
A $triangulation$ is an embedding of a graph on surfaces where every face has length three. In this article, we show the existence of contractible Hamiltonian cycle in triangulated maps of which minimum degree is four.
Barnette and Edelson have shown that there are finitely many minimal triangulations of a connected compact 2-manifold M. Similar finiteness results are obtained for cellular partial triangulations that satisfy various girth inequality…
For a field $\mathbb{F}$, the notion of $\mathbb{F}$-tightness of simplicial complexes was introduced by K\"uhnel. K\"uhnel and Lutz conjectured that any $\mathbb{F}$-tight triangulation of a closed manifold is the most economic of all…
For a tensor triangulated category and any regular cardinal $\alpha$ we study the frame of $\alpha$-localizing tensor ideals and its associated space of points. For a well-generated category and its frame of localizing tensor ideals we…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
This article is a continuation of work on construction and calculation various of modifications of invariant based on the use Euclidean metric values attributed to elements of manifold triangulation. We again address the well investigated…