scholarly journals Balanced Vertex Decomposable Simplicial Complexes and their h-vectors

10.37236/2552 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jennifer Biermann ◽  
Adam Van Tuyl
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Finite Simplicial Complex ◽  
Vertex Decomposable ◽  
Flag Complexes ◽  
Special Case

Given any finite simplicial complex $\Delta$, we show how to construct from a colouring $\chi$ of $\Delta$ a new simplicial complex $\Delta_{\chi}$ that is balanced and vertex decomposable. In addition, the $h$-vector of $\Delta_{\chi}$ is precisely the $f$-vector of $\Delta$.  Our construction generalizes the "whiskering'' construction of Villarreal, and Cook and Nagel. We also reverse this construction to prove a special case of a conjecture of Cook and Nagel, and Constantinescu and Varbaro on the $h$-vectors of flag complexes.

scholarly journals Simplicial Complexes of Whisker Type

10.37236/4894 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Mina Bigdeli ◽  
Jürgen Herzog ◽  
Takayuki Hibi ◽  
Antonio Macchia
Keyword(s):  
Simplicial Complex ◽  
Short Proof ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Linear Resolution ◽  
Cw Complex ◽  
Alexander Dual ◽  
Vertex Decomposable ◽  
The Ideal

Let $I\subset K[x_1,\ldots,x_n]$ be  a zero-dimensional monomial ideal, and $\Delta(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $\Delta(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that  $\Delta(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $\Delta(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.

scholarly journals Simplicial Complexes are Game Complexes

10.37236/6958 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Sara Faridi ◽  
Svenja Huntemann ◽  
Richard J. Nowakowski
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Combinatorial Games ◽  
Natural Question ◽  
Flag Complex ◽  
Game Trees ◽  
Game Value ◽  
Flag Complexes

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known parameters of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.

scholarly journals Maximal Fillings of Moon Polyominoes, Simplicial Complexes, and Schubert Polynomials

10.37236/1167 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Luis Serrano ◽  
Christian Stump
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Canonical Connection ◽  
Cyclic Sieving Phenomenon ◽  
North East ◽  
Schubert Polynomials ◽  
Dyck Paths ◽  
Vertex Decomposable ◽  
Cyclic Sieving

We exhibit a canonical connection between maximal $(0,1)$-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we  show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials. Moreover, for Ferrers shapes we construct a bijection to maximal fillings avoiding south-east chains of the same length which specializes to a bijection between $k$-triangulations of the $n$-gon and $k$-fans of Dyck paths of length $2(n-2k)$. Using this, we translate a conjectured cyclic sieving phenomenon for $k$-triangulations with rotation to the language of $k$-flagged tableaux with promotion.

scholarly journals Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

2017 ◽  
Vol 60 (3) ◽  
pp. 470-477 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo ◽  
Daniel Tanré
Keyword(s):  
Simplicial Complex ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simplicial Complexes ◽  
Connected Components ◽  
Homotopy Groups ◽  
Graded Lie Algebras ◽  
Simplicial Sets ◽  
Graded Lie Algebra ◽  
Finite Simplicial Complex

AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Arithmetic Invariants of Simplicial Complexes

1980 ◽  
Vol 32 (6) ◽  
pp. 1306-1310
Author(s):  
M. Brown ◽  
A. G. Wasserman
Keyword(s):  
Simplicial Complex ◽  
Linear Function ◽  
Euler Characteristic ◽  
Simplicial Complexes ◽  
Homotopy Invariant ◽  
Finite Simplicial Complex ◽  
Arithmetic Invariants

What invariants of a finite simplicial complex K can be computed solely from the values v0(K), V1(K), …, vi(K), … where Vi(K) is the number of i-simplexes of K? The Euler chracteristic χ(K) = Σ i (– 1)ivi(K) is a subdivision invariant and a homotopy invariant while the dimension of K is a subdivision invariant and homeomorphism invariant. In [3], Wall has shown that the Euler chracteristic is the only linear function to the integers that is a subdivision invariant. In this paper we show that the only subdivision invariants (linear or not) of K are the Euler characteristic and the dimension. More precisely we prove the following theorem.

Injectivity of the Connecting Maps in AH Inductive Limit Systems

2005 ◽  
Vol 48 (1) ◽  
pp. 50-68 ◽  
Author(s):  
George A. Elliott ◽  
Guihua Gong ◽  
Liangqing Li
Keyword(s):  
Simplicial Complex ◽  
Inductive Limit ◽  
Extra Condition ◽  
The Third ◽  
Finite Simplicial Complex ◽  
Image Position ◽  
Metrizable Spaces ◽  
Special Case

AbstractLet A be the inductive limit of a systemwith , where Xn,i is a finite simplicial complex, and Pn,i is a projection in M[n,i](C(Xn,i)). In this paper, we will prove that A can be written as another inductive limitwith , where Yn,i is a finite simplicial complex, and Qn, i is a projection inM{n,i}(C(Yn,i)), with the extra condition that all the maps ψn,n+1 are injective. (The result is trivial if one allows the spaces Yn,i to be arbitrary compact metrizable spaces.) This result is important for the classification of simple AH algebras. The special case that the spaces Xn,iare graphs is due to the third author.

scholarly journals EFFICIENT DATA STRUCTURE FOR REPRESENTING AND SIMPLIFYING SIMPLICIAL COMPLEXES IN HIGH DIMENSIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 279-303 ◽  
Author(s):  
DOMINIQUE ATTALI ◽  
ANDRÉ LIEUTIER ◽  
DAVID SALINAS
Keyword(s):  
Data Structure ◽  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Nice Property ◽  
High Dimensions ◽  
Primary Interest ◽  
Flag Complex ◽  
Edge Contraction ◽  
Efficient Data ◽  
Flag Complexes

We study the simplification of simplicial complexes by repeated edge contractions. First, we extend to arbitrary simplicial complexes the statement that edges satisfying the link condition can be contracted while preserving the homotopy type. Our primary interest is to simplify flag complexes such as Rips complexes for which it was proved recently that they can provide topologically correct reconstructions of shapes. Flag complexes (sometimes called clique complexes) enjoy the nice property of being completely determined by the graph of their edges. But, as we simplify a flag complex by repeated edge contractions, the property that it is a flag complex is likely to be lost. Our second contribution is to propose a new representation for simplicial complexes particularly well adapted for complexes close to flag complexes. The idea is to encode a simplicial complex K by the graph G of its edges together with the inclusion-minimal simplices in the set difference Flag (G)\ K. We call these minimal simplices blockers. We prove that the link condition translates nicely in terms of blockers and give formulae for updating our data structure after an edge contraction. Finally, we observe in some simple cases that few blockers appear during the simplification of Rips complexes, demonstrating the efficiency of our representation in this context.

scholarly journals A note on the van der Waerden complex

2019 ◽  
Vol 124 (2) ◽  
pp. 179-187
Author(s):  
Becky Hooper ◽  
Adam Van Tuyl
Keyword(s):  
Simplicial Complex ◽  
Commutative Algebra ◽  
Simplicial Complexes ◽  
Arithmetic Progressions ◽  
Combinatorial Commutative Algebra ◽  
Vertex Decomposable

Ehrenborg, Govindaiah, Park, and Readdy recently introduced the van der Waerden complex, a pure simplicial complex whose facets correspond to arithmetic progressions. Using techniques from combinatorial commutative algebra, we classify when these pure simplicial complexes are vertex decomposable or not Cohen-Macaulay. As a corollary, we classify the van der Waerden complexes that are shellable.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

10.37236/1245 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Art M. Duval
Keyword(s):  
Simplicial Complex ◽  
Simplicial Complexes ◽  
Pure Case ◽  
Definition Of ◽  
Algebraic Shifting

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

scholarly journals Optimal Decision Trees on Simplicial Complexes

10.37236/1900 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Decision Tree ◽  
Decision Trees ◽  
Simplicial Complex ◽  
Elementary Theory ◽  
Simplicial Complexes ◽  
Optimal Decision ◽  
Property A ◽  
Recursive Definition ◽  
Topological Combinatorics ◽  
Definition Of

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension $i$ with respect to a decision tree on a simplicial complex is greater than or equal to the $i$th reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an "optimal" decision tree such that equality holds for each $i$; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.

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