This paper shows how classical ideas used on knot theory can be encoded in a way that makes it possible to use a computer to calculate ambient isotopy invariants of continuous embeddings \(N \ \hookrightarrow \ M\). The authors describe an algorithm for computing the homology and cohomology of a finite connected CW-complex \(X\) with coefficients in a \({\mathbb Z}\pi_{1}(X)\) module \(A\) when \(A\) is finitely generated over \(\mathbb Z\). As examples of the effectiveness of the algorithm, which is implemented in the Groups, Algorithms, and Programming system GAP (http://www.gap-system.org) and in HAP – the homological homological algebra programming system (http://hamilton.nuigalway.ie/) built on GAP – the authors give two illustrations of the technique. The first shows that degree 2 homology distinguishes the homotopy types of the complements of the spun Hopf link and Satoh’s tube map of the welded Hopf links. The second example shows that the system distinguishes between the homeomorphism types of the complements of the granny knot and the reef knot. The details of the implementations are given in the paper. In order to make the computations manageable, a CW complex \(X\) is represented by a regular CW complex \(Y\) – that is one whose attaching maps restrict to homeomorphisms on cell boundaries – together with a simple homotopy equivalence \(Y \simeq X\). The paper contains many useful diagrams to illustrate the constructions used. Timings for the execution of the various codes are given based on execution on a standard GNU/Linux box. Note that GAP/HAP has a substantial number of built-in routines so that the algorithms illustrated in the paper are quite short.