Some Results in Computational Topology

It is the object of this paper to study the topological properties of finite graphs that can be embedded in the <italic>n</italic>-dimensional integral lattice (denoted <italic>N<supscrpt>n</supscrpt></italic>). The basic notion of deletability of a node is first introduced. A node is deletable with respect to a graph if certain computable conditions are satisfied on its neighborhood. An equivalence relation on graphs called reducibility and denoted by “∼” is then defined in terms of deletability, and it is shown that (a) most important topological properties of the graph (homotogy, homology, and cohomology groups) are ∼-invariants; (b) for graphs embedded in <italic>N</italic><supscrpt>3</supscrpt>, different knot types belong to different ∼-equivalence classes; (c) it is decidable whether two graphs are reducible to each other in <italic>N</italic><supscrpt>2</supscrpt> but this problem is undecidable in <italic>N<supscrpt>n</supscrpt></italic> for <italic>n</italic> ≥ 4. Finally, it is shown that two different methods of approximating an <italic>n</italic>-dimensional closed manifold with boundary by a graph of the type studied in this paper lead to graphs whose corresponding homology groups are isomorphic.