Glossary¶
- basis¶
The 0-simplices that lie at the “base” of a simplex. A simplex of order k has (k + 1) 0-simplices in its basis.
- Betti numbers¶
The count of the number of holes of a given dimension in a complex, computed using homology. The 0th Betti number is the number of disconnected components in the complex; the 1st Betti number counts the number of unfilled loops of edges; the 2nd the number of unfilled voids; and so on.
- boundary¶
The set of faces of a simplex.
- boundary operator¶
A linear operator taking k-simplices to the their (k - 1)-simplex faces. The boundary operator extends this to a set (or p-chain) of k-simplices, where it returns all the (k - 1)-simplices in the total boundary.
- closure¶
The closure of a simplex is the set consisting of the simplex and all its component faces, their faces, and so on to its basis.
- complex¶
A collection of simplices. In
simplicial, complexes are closed, in the sense that they contain every face of every simplex contained in the complex.- embedding¶
A geometry imposed on a topological space. Typically an embedding locates each 0-simplex at some point in a Euclidean space, and maps 1- and 2-simplices to lines, surfaces, and so on for higher dimensions. In simplicial topology it is often assumed that only the 0-simplices are located, with the locations of higher simplices being constructed linearly from their bases – so a 1-simplex is a straight line between its endpoints, and so forth.
- Euler characteristic¶
A topological invariant of a complex computed from the alternating sum of the numbers of simplices of different orders:
\[\chi(S) = \sum_{k = 0}^{\infty} (-1)^k \, \#S_k.\]The Euler characteristic is a sort of hole detector for simplicial complexes, in that “un-filled” spaces are counted. For a stronger and more sophisticated (but more computationally demanding) hole detector, use homology.
- face¶
A simplex that lies on the boundary of another simplex. By definition each face has is of order one less than the simplex of which it is a face: 2-simplices (triangle) have faces that are 1-simplices (edges).
- filtration¶
A sequence of simplicial complexes indexed by an ordered index set with the constraint that whenever two indices \(i \le j\) the complexes associated with them are related by inclusion \(C_i \le C_j\). We typically treat the index set as time and the filtration as representing a complex that is built up progressively over time by adding more simplices.
The canonical example of a filtration is the Vietoris-Rips complex.
- flag complex¶
A flag complex is a complex with all “implied” simplices present. That is to say, if all the (k + 1) faces of a k-simplex are present in the complex, then so is the k-simplex itself. Another way to think of this is that all possible triangles of three edges are filled, as are all possible tetrahedra of four triangles, and so forth for higher orders.
- homology¶
A topological invariant of a complex that has a subtle ability to measure holes of different dimensions in a structure.
- order¶
The “dimensionality” of a simplex, A simplex of order 1 (a 1-simplex) is a one-dimensional structure (an edge); a simplex of order 2 is a two-dimensional structure (a triangle); and so on.
- p-chain¶
In homology, a set of p-simplices. The boundary operator is defined on p-chains.
- simplex¶
A component of a complex. A simplex has an order that defines the number of 0-simplices in its basis.
- star¶
The set of simplices of which a given simplex is a part. The star is not necessarily a closed simplicial complex, but the star of the closure (or indeed the closure of the star) is (and they are generally different).
- topology¶
“The stratosphere of human thought! In the twenty-fourth century it might possibly be of use to someone…” (Alexander Solzhenitsyn, The First Circle).
- topological invariant¶
A property that isn’t changed by smooth changes in a complex, or by the details of how a shape is approximated by a complex.
- Vietoris-Rips complex¶
A complex derived from an underlying distance metric. If 0-simplices are given locations, then for a given distance eps the Vietoris-Rips complex at scale \(\epsilon\) has a k-simplex for every collection of (k + 1) 0-simplices lying mutually within a distance \(\epsilon\) of each other.
Another of saying this is that the Vietoris-Rips complex is the flag complex resulting from a complex consisting of 1-simplices between all pairs of 0-simplices lying within a distance \(\epsilon\) of each other.
If we take a set of geometrically located 0-simplices allow \(\epsilon\) to increase, the resulting sequence of complexes forms a filtration.