  We show how the classical notions of cohomology with local coefficients,
CW-complex, covering space, homeomorphism equivalence, simple homotopy
equivalence, tubular neighbourhood, and spinning can be encoded on a computer
and used to calculate ambient isotopy invariants of continuous embeddings
$N\hookrightarrow M$ of one topological manifold into another. More
specifically, we describe an algorithm for computing the homology $H_n(X,A)$
and cohomology $H^n(X,A)$ of a finite connected CW-complex X with local
coefficients in a $\mathbb Z\pi_1X$-module $A$ when $A$ is finitely generated
over $\mathbb Z$. It can be used, in particular, to compute the integral
cohomology $H^n(\widetilde X_H,\mathbb Z)$ and induced homomorphism
$H^n(X,\mathbb Z) \rightarrow H^n(\widetilde X_H,\mathbb Z)$ for the covering
map $p\colon \widetilde X_H \rightarrow X$ associated to a finite index
subgroup $H < \pi_1X$, as well as the corresponding homology homomorphism.
  We illustrate an open-source implementation of the algorithm by using it to
show that: (i) the degree $2$ homology group $H_2(\widetilde X_H,\mathbb Z)$
distinguishes between the homotopy types of the complements $X\subset \mathbb
R^4$ of the spun Hopf link and Satoh's tube map of the welded Hopf link (these
two complements having isomorphic fundamental groups and integral homology);
(ii) the degree $1$ homology homomorphism $H_1(p^{-1}(B),\mathbb Z) \rightarrow
H_1(\widetilde X_H,\mathbb Z)$ distinguishes between the homeomorphism types of
the complements $X\subset \mathbb R^3$ of the granny knot and the reef knot,
where $B\subset X$ is the knot boundary (these two complements again having
isomorphic fundamental groups and integral homology).
