Computational topology develops algorithms for topological questions about finite representations of spaces, such as simplicial complexes. It connects discrete data structures to invariants that ignore irrelevant geometric distortion. connected components and homology groups are examples of computable structural information.[ref]
For a small simplicial homology calculation over a field, take three vertices joined by three edges in a cycle, without a filled face. The matrix of the first boundary operator has matrix rank two, so this linear map has a kernel of a linear map of dimension one; with no two-dimensional face, that cycle is not a boundary. Thus the first Betti number is one. Adding the filled triangular face makes the same cycle a boundary and reduces the first Betti number to zero. The vertex coordinates need not change: the encoded incidence structure determines whether the hole is present.
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Computational topology is a branch of mathematics and computer science that focuses on the study of topological properties and structures through computational methods. It combines techniques from topology, a field concerned with the properties of space that are preserved under continuous transformations, with algorithms and data structures to analyze and manipulate topological spaces.