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Type of mathematical set
In mathematics, a simplicial complex is a structured set of simplices (for example, points, line segments, triangles, and their n-dimensional counterparts)
Simplicial_complex
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local
Link_(simplicial_complex)
Mathematical object
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking
Abstract_simplicial_complex
Abstract simplicial complex describing a graph's cliques
subgraphs) of an undirected graph. The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed
Clique_complex
Representation of mathematical space
triangulation describes the replacement of topological spaces with simplicial complexes by the choice of an appropriate homeomorphism. A space that admits
Triangulation_(topology)
Topological data
tree is a type of trie used to efficiently represent any general simplicial complex. Through its nodes, this data structure notably represents all the
Simplex_tree
Topics referred to by the same term
simplicial complex Link (knot theory), a collection of knots entangled with one another Link function in statistics Link (Mars), a rock outcrop Link or
Link
Topics referred to by the same term
contexts. See: Geometric realization of an abstract simplicial complex; Geometric realization of a simplicial set. This disambiguation page lists mathematics
Geometric_realization
Complex in algebraic topology
algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which
Čech_complex
Subgraph induced by all nodes linked to a given node of a graph
a related concept in polyhedra Link (simplicial complex), a generalization of the neighborhood to simplicial complexes Hell 1978, Sedláček 1983 Wigderson
Neighbourhood_(graph_theory)
Topics referred to by the same term
sets Chain complex, an algebraic structure Simplicial complex, a kind of topological space CW complex, a kind of topological space Line complex, a 3-dimensional
Complex
Endofunctor on the category of simplicial sets
of simplicial sets (subdivision functor or Sd functor) is an endofunctor on the category of simplicial sets. It refines the structure of simplicial sets
Subdivision_(simplicial_set)
Mathematical ring
Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic
Stanley–Reisner_ring
Algebraic structure associated with a topological space
the task easier. The simplicial homology groups Hn(X) of a simplicial complex X are defined using the simplicial chain complex C(X), with Cn(X) the free
Homology_(mathematics)
Research field in deep learning
interactions among multiple entities and complex hierarchies. This approach leverages structures like simplicial complexes and hypergraphs to capture global
Topological_deep_learning
Generalized manifold
i of X ', a simplicial complex Li' endowed with a rigid simplicial action of a finite group Γi. a simplicial map φi of Li' onto the link Li of i in X
Orbifold
Branch of mathematics
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Algebraic_topology
to simplicial complexes and CW complexes in the computation of the homology of topological spaces. Non-positively curved and CAT(0) cube complexes appear
Cubical_complex
Branch of geometry that studies combinatorial properties and constructive methods
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory
Discrete_geometry
Topics referred to by the same term
n-dimensional analogue of a triangle Simplicial polytope, a polytope with all simplex facets Simplicial complex, a collection of simplicies Pascal's simplex
Simplex_(disambiguation)
Mathematical structure
conditions on the complexes Δ that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at
Building_(mathematics)
Approximation to shape of a point cloud
α-complex of the given set of points is the simplicial complex formed by the set of edges and triangles whose radii are at most 1/α. The α-complex is
Alpha_shape
In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed
Curve_complex
All points in the topological closure not belonging to the interior
slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is
Boundary_(topology)
Mathematical object in topological graph theory
A chessboard complex is a particular kind of abstract simplicial complex, which has various applications in topological graph theory and algebraic topology
Chessboard_complex
Mathematical group of the homotopy classes of loops in a topological space
space of a finite connected simplicial complex X {\displaystyle X} can also be described directly as a simplicial complex using edge-paths. Its vertices
Fundamental_group
Topics referred to by the same term
graph, whereby replacing the edges by paths Subdivision (simplicial complex) Subdivision (simplicial set) Subdivision surface, in computer graphics Subdivision
Subdivision
topological data analysis (TDA) to associate a sequence of nested simplicial complexes to a finite data set in order to detect the persistence of topological
Degree-Rips_bifiltration
Topics referred to by the same term
System In geometry, Barycentric subdivision, a way of dividing a simplicial complex Barycentric coordinates (mathematics), coordinates defined by the
Barycentric
Branch of topology
since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from Rn. The Zariski topology is defined
General_topology
Any collection of sets, or subsets of a set
sets as its vertices. An abstract simplicial complex is a combinatorial abstraction of the notion of a simplicial complex, a shape formed by unions of line
Family_of_sets
Topological manifold whose homology coincides with that of a sphere
example of a finite simplicial complex that is a topological manifold but not a PL manifold. (It is not a PL manifold because the link of a point is not
Homology_sphere
Probability distribution
normalizes generalized gamma variates, one obtains variates from the simplicial generalized beta distribution (SGB). On the other hand, SGB variates can
Dirichlet_distribution
Branch of mathematics
data points with a family of simplicial complexes, indexed by a proximity parameter. Analyse these topological complexes via algebraic topology – specifically
Topology
Feature of a polyhedron, polytope, etc.
facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex. For (boundary complexes of) simplicial
Facet_(geometry)
Mathematical theory
L)&\cong 0\\\end{aligned}}} Let X {\displaystyle X} be an abstract simplicial complex on a vertex set V {\displaystyle V} of size n {\displaystyle n} .
Alexander_duality
means that X is a non-branching simplicial complex. Condition 3 means that X is a strongly connected simplicial complex. If we require Condition 2 to hold
Pseudomanifold
Topics referred to by the same term
abstract cell complex Cell, a basic unit of a cellular automaton Cell, an element of a CW complex Cell, a k-face of a simplicial complex Cell (journal)
Cell
Structure from which the geometry of the universe arises
geometric simplicial complex and then geometric simplices are stitched together into a piecewise linear space. Developed further, triangulation, link distance
Pregeometry_(physics)
refer to such a construction as a simplicial complex. The boundary operator on this triangulation/simplicial complex T is defined in the usual way: for
Discrete_exterior_calculus
{\displaystyle X_{i}\setminus X_{i-1}} . Examples: If X is an n-dimensional simplicial complex such that every simplex is contained in an n-simplex and n−1 simplex
Intersection_homology
Generalization of category theory
a new object of study in category theory. Weak Kan complexes, or quasi-categories, are simplicial sets satisfying a weak version of the Kan condition
Higher_category_theory
Regular tiling of the plane
triangular tiling is called an A2 lattice. It is the 2-dimensional case of a simplicial honeycomb. The A* 2 lattice (also called A3 2) can be constructed by the
Triangular_tiling
American mathematician (1933–2024)
Simplicial homotopy theory. Curtis died in Seattle, Washington on April 2, 2024, at the age of 91. The Lower Central Series for Free Group Complexes.
Edward_B._Curtis
Topics referred to by the same term
Whitehead group of the group ring. The Whitehead group Wh(A) of a simplicial complex or PL-manifold A, equal to Wh(π1(A)); see Whitehead torsion. All named
Whitehead_group
Tree graph with one central node and leaves of length 1
subdivision of a star Star (simplicial complex) - a generalization of the concept of a star from a graph to an arbitrary simplicial complex. Wikimedia Commons
Star_(graph_theory)
Conjecture on zeros of the zeta function
equivalent to the statement that the Euler characteristic of the simplicial complex determined by the lattice of integers under divisibility is o ( n
Riemann_hypothesis
Natural number
The barycentric subdivision of a tetrahedron produces an abstract simplicial complex with exactly 149 simplices. Sloane, N. J. A. (ed.). "Sequence A001632
149_(number)
Topological manifold with a piecewise linear structure on it
a manifold. It usually means a piecewise linear manifold made by simplicial complexes. A digital manifold is a special kind of combinatorial manifold which
Piecewise_linear_manifold
Generalization of the notion of convergence that is found in general topology
Retrieved 14 January 2021.{{cite journal}}: CS1 maint: periodical has ISBN (link) Dolecki, Szymon; Mynard, Frédéric (2014). "A unified theory of function
Convergence_space
Non-orientable surface with one edge
from an abstract simplicial complex, because all three triangles share the same three vertices, while abstract simplicial complexes require each triangle
Möbius_strip
Class of problems
extended beyond these planar environments to handle complex 3D terrains represented as simplicial complexes or triangular meshes. This advancement is particularly
Kinodynamic_planning
Subdivision of the plane by lines
There are three known infinite families of simplicial arrangements, as well as many sporadic simplicial arrangements that do not fit into any known family
Arrangement_of_lines
Technique in topological data analysis
point cloud, a graph, a function, or, more generally, a simplicial complex or a chain complex. Generally, longer intervals in a barcode correspond to
Persistence_barcode
category D. Set, the category of (small) sets. sSet, the category of simplicial sets. "weak" instead of "strict" is given the default status; e.g., "n-category"
Glossary_of_category_theory
Subdivision of space into cells
discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells
Mesh_generation
Tool to track locally defined data attached to the open sets of a topological space
Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering. 1938 Hassler Whitney gives a 'modern' definition
Sheaf_(mathematics)
Branch of topology
homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem
Low-dimensional_topology
Type theory in logic and mathematics
products, sums and universes and of a model of this type theory in Kan simplicial sets. It began by saying "The homotopy λ-calculus is a hypothetical (at
Homotopy_type_theory
Combinatorial representation of a graph on an orientable surface
and processing, in geometrical modeling. This model is related to simplicial complexes and to combinatorial topology. A combinatorial map is a boundary
Combinatorial_map
American mathematician (born 1931)
In 1961, Milnor disproved the Hauptvermutung by illustrating two simplicial complexes that are homeomorphic but combinatorially distinct, using the concept
John_Milnor
Combinatorial sequence of numbers
{\displaystyle n} generators, and one more than the number of abstract simplicial complexes on a set with n {\displaystyle n} elements. Accurate asymptotic estimates
Dedekind_number
Finest topology making some functions continuous
second-countable Homotopy homotopy group fundamental group Simplicial complex CW complex Polyhedral complex Manifold topological smooth Bundle (mathematics) Cobordism
Final_topology
Type of mathematical function
affine space, as well as on piecewise linear manifolds and simplicial complexes (see simplicial map). In each case, the function may be real-valued, or it
Piecewise_linear_function
Topics referred to by the same term
for several species of moth n-skeleton, the subcomplex of a simplicial complex or CW complex consisting of all faces of or below a certain dimension Skeleton
Skeleton_(disambiguation)
Analysis of datasets using techniques from topology
simplicial complex to a much smaller cellular complex which is homotopic to the original one. This reduction can in fact be performed as the complex is
Topological_data_analysis
Fourth letter in the Greek alphabet
the position of which is variant between isomeric forms. A simplex, simplicial complex, or convex hull. In chemistry, the addition of heat in a reaction
Delta_(letter)
Two-dimensional manifold
difficult result that every compact 2-manifold is homeomorphic to a simplicial complex, which is of interest in its own right. The most common proof of the
Surface_(topology)
American mathematician
wiring diagram syntax. His unpublished work "Metric realization of fuzzy simplicial sets" was cited by the authors of UMAP as inspirational for that work
David_Spivak
Theoretical object in mathematics
abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an n‑dimensional abstract simplicial complex, and
Field_with_one_element
Links the homology groups of a product space with those of the individual spaces
(The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex C ∗ ( X ) ⊗ C ∗ ( Y ) {\displaystyle
Eilenberg–Zilber_theorem
1946, it asks, given an integral homology class in degree n of a simplicial complex, is it the image by a continuous mapping of the fundamental class
Timeline_of_bordism
g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Topics referred to by the same term
Taiwan Assets Scrutiny Committee, former government agency Abstract simplicial complex Apoptosis-associated speck-like protein containing a CARD, a protein
ASC
Topics referred to by the same term
that resembles a star Star (simplicial complex), the set of simplices containing a given vertex in a simplicial complex Kleene star, which can refer
Star_(disambiguation)
Dutch mathematician and logician
reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings. In 1912, at age
L._E._J._Brouwer
Mathematical space
manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex. If the form is Z {\displaystyle \mathbb {Z} } , there are two manifolds
4-manifold
In mathematics, a non-algebraic number
S2CID 122023355. Heuer, Nicolaus; Loeh, Clara (1 November 2019). "Transcendental simplicial volumes". arXiv:1911.06386 [math.GT]. "Real number". Encyclopædia Britannica
Transcendental_number
Class of artificial neural networks
More powerful GNNs operating on higher-dimension geometries such as simplicial complexes can be designed. As of 2022[update], whether or not future architectures
Graph_neural_network
analysis, machine learning, and physics. Let K {\displaystyle K} be a simplicial complex, and let f : K → R {\displaystyle f:K\to \mathbb {R} } be a monotonic
Persistent_Betti_number
Convex hull of a finite set of points in a Euclidean space
as a spherical tiling. A convex polytope can be decomposed into a simplicial complex, or union of simplices, satisfying certain properties. Given a convex
Convex_polytope
Triangulation method
developed. Typically, the domain to be meshed is specified as a coarse simplicial complex; for the mesh to be numerically stable, it must be refined, for instance
Delaunay_triangulation
Surface that locally minimizes its area
discrete differential geometry discrete minimal surfaces are studied: simplicial complexes of triangles that minimize their area under small perturbations of
Minimal_surface
Croatian-American mathematician
(1988), no. 380 Bestvina, Mladen; Feighn, Mark, Bounding the complexity of simplicial group actions on trees. Inventiones Mathematicae, vol. 103 (1991), no
Mladen_Bestvina
Vertices connected in pairs by edges
graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of
Graph_(discrete_mathematics)
Area of discrete mathematics
graph in a topology is a set of simplexes that is called the simplicial one-dimensional complex. This subarea studies the embedding (or imbedding) of a graph
Graph_theory
Algebra of formal sums
space, for instance as the set of k {\displaystyle k} -simplices in a simplicial complex, or the set of singular k {\displaystyle k} -simplices in a manifold
Free_abelian_group
Algebraic construct classifying topological spaces
model categories. It is possible to define abstract homotopy groups for simplicial sets. Homology groups are similar to homotopy groups in that they can
Homotopy_group
mathematical framework to describe and analyze set systems, or equivalently simplicial complexes. This idea was first introduced by Ronald Atkin in the early 1970s
Q-analysis
Area of combinatorics
Hochster, Melvin (1977). "Cohen–Macaulay rings, combinatorics, and simplicial complexes". Ring Theory II: Proceedings of the Second Oklahoma Conference.
Algebraic_combinatorics
Textbook in topology
topology discussed as part of this presentation include simplicial complexes, fundamental groups, simplicial homology and singular homology, and the Poincaré
A Guide to the Classification Theorem for Compact Surfaces
A_Guide_to_the_Classification_Theorem_for_Compact_Surfaces
Generalization of graph theory
called an abstract simplicial complex. It is generally not reduced, unless all hyperedges have cardinality 1. An abstract simplicial complex with the augmentation
Hypergraph
American mathematician
abstract simplicial complex whose face ring is a Cohen–Macaulay ring can be partitioned into disjoint intervals, each including a facet of the complex. Such
Caroline_Klivans
Algebraic tool for computing topological spaces' invariants
spaces encountered in topology are topological manifolds, simplicial complexes, or CW complexes, which are constructed by piecing together very simple patches
Mayer–Vietoris_sequence
Undirected graph with 11 nodes and 27 edges
ISBN 978-3-031-22562-8 Ewald, Günter (1973), "Hamiltonian circuits in simplicial complexes", Geometriae Dedicata, 2 (1): 115–125, doi:10.1007/BF00149287, S2CID 122755203
Goldner–Harary_graph
Adjacent subset of an undirected graph
involve cliques in graphs. Among them, The clique complex of a graph G is an abstract simplicial complex X(G) with a simplex for every clique in G A simplex
Clique_(graph_theory)
Statistical model used in machine learning
Calibration". arXiv:2408.02841 [stat.ML]. Graf, Monique (2019). "The Simplicial Generalized Beta distribution - R-package SGB and applications". Libra
Flow-based_generative_model
Four-dimensional analog of the octahedron
fourth dimension)." Tyrrell & Semple 1971. Coxeter 1973, p. 130, § 7.6; "simplicial subdivision". Coxeter 1973, pp. 292–293, Table I(ii); "16-cell, 𝛽4".
16-cell
Italian theoretical physicist (1931–2014)
requiring complex angles). This formulation is known as Regge theory. In the early 1960s, Regge introduced Regge calculus, a simplicial formulation
Tullio_Regge
Subfield of mathematical topology
whether two closed, oriented 3-manifolds given by triangulations (simplicial complexes) are equivalent (homeomorphic) is elementary recursive. This generalizes
Computational_topology
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