Irreducible triangulations of low genus surfaces
Thom Sulanke
已知亏格为0、1或2的可定向曲面和亏格为1、2、3或4的不可定向曲面的不可约三角剖分完备集。通过研究这些集合，我们确定了这些不可约三角剖分的一些性质。
The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of these irreducible triangulations.
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Generating irreducible triangulations of surfaces
从一个固定曲面的不可约三角剖分和顶点分裂开始，可以生成该曲面直到给定顶点数的所有三角剖分。曲面S_0、S_1、N_1和N_2的不可约三角剖分已确定。提出了一种利用其它曲面的三角剖分生成固定曲面的不可约三角剖分的算法。该算法已被实现为计算机程序，其终止于S_1、S_2、N_1、N_2、N_3和N_4。因此，对于S_2、N_3和N_4，完全不可约三角剖分集现在也是已知的，其基数分别为396784、9708和6297982。
A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$ vertices. The best previous bound was $171g-72$.
Note on the Irreducible Triangulations of the Klein Bottle
我们给完整的列表的29个不可约三角的克莱因瓶。我们将展示如何建设Lawrencenko和Negami，其中列出只有25个这样的不可约三角，可以修改在两个点，以产生4个额外的不可约三角的克莱因瓶。
A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. In this work we propose an algorithm that constructs the set of irreducible triangulations of any surface with precisely one boundary component. By implementing the algorithm on computer, we have found a list of 297 nonisomorphic combinatorial types of irreducible triangulations on the once-punctured torus.
The separating systole and the genus ratio of high genus triangulations
研究高亏格三角剖分的分离系绳和亏格比的收敛性，揭示其对数阶性质及计数结果。
Large expander subgraphs in high genus triangulations
研究高亏格三角剖分中存在大扩张子图的性质，通过新标准证明其存在性并回答Benjamini问题。
Schnyder woods are a well-known combinatorial structure for plane triangulations, which yields a decomposition into 3 spanning trees. We extend here definitions and algorithms for Schnyder woods to closed orientable surfaces of arbitrary genus. In particular, we describe a method to traverse a triangulation of genus $g$ and compute a so-called $g$-Schnyder wood on the way. As an application, we give a procedure to encode a triangulation of genus $g$ and $n$ vertices in $4n+O(g \log(n))$ bits. This matches the worst-case encoding rate of Edgebreaker in positive genus. All the algorithms present…
Exterior Shifting of Low Genus Surfaces
本研究刻画了环面、射影平面及克莱因瓶所有三角剖分的可能外移位，并提出确定性多项式时间算法以计算这类给定三角剖分的外移位。
A complete list of irreducible triangulations is identified on the Möbius band.
Transversal structures on triangulations: a combinatorial study and straight-line drawings
本文研究了一种特殊的组合结构，这种组合结构是外表面为四边形且无分离三角形的三角化平面图所特有的，称为不可约三角化。这种结构是由何欣以正则边标号的名义提出的，它由两个横截的双极取向组成。因此，这里使用的术语是横向结构。本文的主要结果是不可约三角剖分与三叉树之间的一个双射，以及不可约三角剖分的一个直线作图算法。对于具有$n$顶点的随机不可约三角剖分，图形的网格大小以高概率$11n/27\times 11n/27$渐近，直到添加性误差$\cO(\sqrt{n})$。相比之下，用于这些三角剖分的最佳先前已知算法仅保证网格大小$(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$。
Distances and isoperimetric inequalities in random triangulations of high genus
本研究证明亏格与规模n成正比的均匀随机三角剖分直径高概率下为log n量级、多数点对距离差至多为加性常数，核心工具是具备独立研究价值的等周不等式。
A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boun…
Topological triangulations of orientable and non-orientable surfaces with arbitrary genus have important applications in quantum geometry, graph theory and statistical physics. However, until now only the asymptotics for 2-spheres are known analytically, and exact counts of triangulations are only available for both small genus and small triangulations. We apply the Wang-Landau algorithm to calculate the number $N(m,h)$ of triangulations for several order of magnitudes in system size $m$ and genus $h$. We verify that the limit of the entropy density of triangulations is independent of genus an…
Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4-manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the sim…
Typical distances in high-genus triangulations
研究亏格与面数成比例的均匀随机三角剖分中两点距离，证明距离经对数缩放后依概率收敛到确定常数，证实了Budzinski、Chapuy和Louf的猜想。
Local limits of uniform triangulations in high genus
本研究证明了高亏格均匀随机三角剖分的局部极限为平面随机双曲三角剖分（PSHT），结合组合与概率方法完成论证，并得到相关计数渐近结果与弱马尔可夫三角剖分的结构结论。
Local limits of uniform triangulations with boundaries in high genus
研究高亏格均匀三角剖分的局部极限，证明具有边界时的局部极限为半平面双曲三角剖分，并首次构造了这些结构作为大亏格三角剖分的局部极限。
New bijective links on planar maps via orientations
本文给出了平面映射上新的双射。首先在平面地图上的双极定向与无分隔3圈的4边形三角剖分上的特定“横截结构”之间建立一个双射，称之为不可约三角剖分。这种双射专门用于有根不可分映射和有根不可约三角剖分之间的双射。这反过来又产生了一个双射之间的有根无环地图和有根三角形，根据观察，无环地图和三角形被分解成组件，分别是不可分的地图和不可约的三角形。这给出了另一个双射证明（后Wormald的建设出版于1980年）的事实，即植根loopless地图与$n$边缘是equinumerous植根三角与$n$内顶点。
We prove that any triangulation of a surface different from the sphere and the projective plane admits an orientation without sinks such that every vertex has outdegree divisible by three. This confirms a conjecture of Barát and Thomassen and is a step towards a generalization of Schnyder woods to higher genus surfaces.
We present a family of index 1 abelian current graphs whose derived embeddings can be modified into triangulations of $K_{12s}$ for $s \geq 4$. Our construction is significantly simpler than previous methods for finding genus embeddings of $K_{12s}$, which utilized either large index or nonabelian groups.
A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the once-punctured torus, consisting of exactly 297 non-isomorphic triangulations.
Decomposition and Enumeration of Triangulated Surfaces
我们描述了一些理论成果的三角形的表面，我们开发了一个理论的根，分解和属表面。我们应用这个理论来描述一个算法，列出所有的三角剖分的闭曲面，最多有一个固定的顶点数。我们专门的理论的情况下，顶点的数量是最多11，我们得到理论上的限制，使我们能够得到列表的三角剖分的闭曲面与最多11个顶点的属表面。
Enumerating all 3-manifold triangulations of a given size is a difficult but increasingly important problem in computational topology. A key difficulty for enumeration algorithms is that most combinatorial triangulations must be discarded because they do not represent topological 3-manifolds. In this paper we show how to preempt bad triangulations by detecting genus in partially-constructed vertex links, allowing us to prune the enumeration tree substantially. The key idea is to manipulate the boundary edges surrounding partial vertex links using expected logarithmic time operations. Practical…
A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible triangulations of any punctured surface is established. Complete lists of irreducible triangulations are determined for the Möbius band (6 in number) and the pinched torus (2 in number). All the non-isomorphic combinatorial types (20 in number) of triangulations of the projective plane with up to 8 vertices are determined.
Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}γ^{2n}(2n)!$, where $γ$ is an algebraic constant and $c_g$ is a constant depending only on the genus $g$. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$, $g\ge 1$, has exactly one non-planar component.
Let $Φ$ be an irreducible crystallographic root system and $\mathcal P$ its root polytope, i.e., its convex hull. We provide a uniform construction, for all root types, of a triangulation of the facets of $\mathcal P$. We also prove that, on each orbit of facets under the action of the Weyl gruop, the triangulation is unimodular with respect to a root sublattice that depends on the orbit.
Non-Realizable Minimal Vertex Triangulations of Surfaces: Showing Non-Realizability using Oriented Matroids and Satisfiability Solvers
我们证明了亏格为6的闭连通可定向2-流形的极小顶点三角剖分不允许在R^3中有多面体嵌入。我们还提供了最小顶点三角形的封闭，连通，定向2流形的亏格5，不承认任何多面体嵌入的例子。我们构造了一个新的无限族曲面的不可实现三角剖分。这些结果是通过将寻找合适的定向拟阵的问题转化为可满足性问题来实现的。该方法也可应用于其它几何可实现性问题，如多面体的面格。
On the size of k-irreducible triangulations
研究k-不可约三角剖分的三角数目，证明其为O(k²g)，改进了之前的结果。
Counting 1-vertex Triangulations Of Oriented Surfaces
有向紧致曲面$S$的{PDMATHABMMPDMATHABMN顶点三角剖分}是一个嵌入图$T\subset S$，它具有唯一顶点，使得$S\setminus T$的所有连通分支都是三角形（恰好与$T$的3条边相邻）.本文给出了在给定亏格的有向曲面上的这类三角剖分（直到等价）的计数公式。{ \em Une triangulation à un sommet} d'une surface orientée proximete $S$ de genre $g$ est un graphe $T\subset S$ qui a un unique sommet et dont toutes les faces（composite connexes de $S\setminus T$）sont des triangles（incidentes à trois arêtes de $T$）. Cet article donne des formules permettant d'énumérer ces triangulations.
It is well known that any triangulation of a marked surface produces a quiver. In this paper we will provide a triangulation for orientable surfaces of genus $n$ with an arbitrary number interior marked points (called punctures) whose corresponding quiver has a maximal green sequence.
Motivated by recent work on Delaunay triangulations of hyperbolic surfaces, we consider the minimal number of vertices of such triangulations. First, we will show that every hyperbolic surface of genus $g$ has a simplicial Delaunay triangulation with $O(g)$ vertices, where edges are given by distance paths. Then, we will construct a class of hyperbolic surfaces for which the order of this bound is optimal. Finally, to give a general lower bound, we will show that the $Ω(\sqrt{g})$ lower bound for the number of vertices of a simplicial triangulation of a topological surface of genus $g$ is tigh…
We tackle the classification problem of non-degenerate potentials for quivers arising from triangulations of surfaces in the cases left open by Geiss-Labardini-Schröer. Namely, for once-punctured closed surfaces of positive genus, we show that the quiver of any triangulation admits infinitely many non-degenerate potentials that are pairwise not weakly right-equivalent; we do so by showing that the potentials obtained by adding the 3-cycles coming from triangles and a fixed power of the cycle surrounding the puncture are well behaved under flips and QP-mutations. For twice-punctured closed surf…
Polyhedra of genus 2 with 10 vertices and minimal coordinates
给出了4 × 4 × 4立方体中亏格2的可定向曲面的全部865个顶点极小三角剖分在一般位置上的坐标极小几何实现。
We determine some of the derived equivalences of a class of gentle algebras called surface algebras. These algebras are constructed from an unpunctured Riemann surface of genus 0 with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. In particular, we fix a triangulation of a surface and determine when different cuts produce derived equivalent algebras.
Polyhedra of genus 3 with 10 vertices and minimal coordinates
对5 × 5 × 5立方体中亏格为3的可定向曲面的20个顶点最小三角剖分中的17个，给出了坐标最小几何实现.
The problems of the genus of the complete graphs and minimum triangulations for each surface were both solved using the theory of current graphs, and each of them divided into twelve different cases, depending on the residue modulo 12 of the number of vertices. Cases 8 and 11 were of particular difficulty for both problems, with multiple families of current graphs developed to solve these cases. We solve these cases in a unified manner with families of current graphs applicable to both problems. Additionally, we give new constructions to both problems for Cases 6 and 9, which greatly simplify …
Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds
本文给出了组合2-流形和组合3-流形的一个快速计数算法。特别地，我们列举了所有的三角形曲面与11和12顶点和所有的三角形3-流形与11顶点。进一步确定了亏格4的不可定向曲面上的所有等值线多面体映射，亏格3的可定向曲面和亏格5、6的不可可定向曲面的所有等值线三角剖分。
It is known that for every $α\geq 1$ there is a planar triangulation in which every ball of radius $r$ has size $Θ(r^α)$. We prove that for $α<2$ every such triangulation is quasi-isometric to a tree. The result extends to Riemannian 2-manifolds of finite genus, and to large-scale-simply-connected graphs. We also prove that every planar triangulation of asymptotic dimension 1 is quasi-isometric to a tree.
Higher chromatic numbers $χ_s$ of simplicial complexes naturally generalize the chromatic number $χ_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $χ_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $χ_{d+1}=1$, and only little is known on $χ_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see …
Minimal Simplicial Degree $d$ Maps from Genus $g$ Surfaces to the Torus
本文研究了从亏格 g 曲面到环面的最小 simplicial 度数 d 映射，针对不同亏格情况提出了极小构造及开放问题。
Generalized break divisors and triangulations of Lawrence polytopes
本文利用余图拟阵 Lawrence 多面体的三角剖分定义生成树上的函数，构造图的 g 次 Picard 群代表元，并刻画其与结点曲线稳定性条件的对应关系。
Partial triangulations of surfaces with girth constraints
本文研究曲面上带围长约束的部分三角剖分与稀疏嵌入图，利用高亏格围长不等式刻画相关图类，并证明收缩极小的 (3,6)-紧或 (3,3)-紧 M-嵌入图具有有限性。
Congruence classes of monodromies of even triangulations
本文计数一般闭曲面上偶三角剖分的单值群同余类数量，此前已明确球面、射影平面、环面的对应数量，该研究拓展了低亏格曲面的相关结论。
In this paper, the quadrangular genus (4-genus) of the complete graph $K_p$ is shown to be $γ_4 (K_p) = \lceil {p(p-5)}/{8} \rceil +1$ for orientable surfaces. This means that $K_p$ is minimally embeddable in the closed orientable surface of genus $γ_4 (K_p)$ under the constraint that each face has length at least 4. In the most general setting, the genus of the complete graph was established by Ringel and Youngs and was mainly concerned with triangulations of surfaces. Nonetheless, since then a great deal of interest has also been generated in quadrangulations of surfaces. Hartsfield and Ring…
Given a marked surface (S,M) we can add arcs to the surface to create a triangulation, T, of that surface. For each triangulation, T, we can associate a cluster algebra. In this paper we will consider the torus of genus n with two interior marked points (called punctures). We will construct a specific triangulation of this surface which yeilds a specific quiver. Then in the sense of work by Keller we will produce a maximal green sequence for this quiver.
We prove that for every surface $Σ$ of Euler genus $g$, every edge-maximal embedding of a graph in $Σ$ is at most $O(g)$ edges short of a triangulation of $Σ$. This provides the first answer to an open problem of Kainen (1974).
We present a procedure to sample uniformly from the set of combinatorial isomorphism types of balanced triangulations of surfaces - also known as graph-encoded surfaces. For a given number $n$, the sample is a weighted set of graph-encoded surfaces with $2n$ triangles. The sampling procedure relies on connections between graph-encoded surfaces and permutations, and basic properties of the symmetric group. We implement our method and present a number of experimental findings based on the analysis of $138$ million runs of our sampling procedure, producing graph-encoded surfaces with up to $280$ …
Motivated by fixed-parameter tractable (FPT) problems in computational topology, we consider the treewidth of a compact, connected 3-manifold $M$ defined by \[ \operatorname{tw}(M) = \min\{\operatorname{tw}(Γ(\mathcal{T})):\mathcal{T}~\text{is a triangulation of }M\}, \] where $Γ(\mathcal{T})$ denotes the dual graph of $\mathcal{T}$. In this setting the relationship between the topology of a 3-manifold and its treewidth is of particular interest. First, as a corollary of work of Jaco and Rubinstein, we prove that for any closed, orientable 3-manifold $M$ the treewidth $\operatorname{tw}(M)$ is…
We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess …
Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1
我们推导出了所有亏格为1的Hurwitz数的封闭表达式，并给出了亏格为0和1的Hurwitz数的一个简单的新的图论解释。（赫维茨数本质上计算球面的不可约亏格g覆盖，在一个点上有任意指定的分支，在其他指定点上有简单分支，没有其他分支。这个问题等价于将置换的传递因式分解计数为置换。）这些结果证明了Goulden和Jackson的一个猜想，推广了Hurwitz等人的结果.
The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar.
We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus $g$ with a single boundary curve and $n$ marked points on this curve; we consider triangulations up to homeomorphism with the marked points as their vertices. Our main results are upper and lower bounds on the maximal distance between triangulations depending on $n$ and can be thought of as bounds on the diameter of flip-graphs up to the quotient of underlying homeomorphism groups. The ma…
The Sato-Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this paper, we explicitly compute the auto-correlation functions of Sato-Tate distributions for genus 2 curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups $Sp(2m)$ for all $m \in \mathbb Z_{\ge 1}$, which have interest in their own rights.
We consider maps on a surface of genus $g$ with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus $g$ and fixed number $n$ of faces with circumferences $α_1,\ldots,α_n$ and a $β$-irreducibility constraint, which roughly requires that all contractible cycles have length at least $β$. Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume $V_{g,n}^{(β)}(α_1,\ldots,α_n)$ of this family of maps that arises naturally from the Lebesgu…
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