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We show how the classical notions of cohomology with local coefficients,CW-complex, covering space, homeomorphism equivalence, simple homotopyequivalence, tubular neighbourhood, and spinning can be encoded on a computerand used to calculate ambient isotopy invariants of continuous embeddings$N\hookrightarrow M$ of one topological manifold into another. Morespecifically, 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 localcoefficients in a $\mathbb Z\pi_1X$-module $A$ when $A$ is finitely generatedover $\mathbb Z$. It can be used, in particular, to compute the integralcohomology $H^n(\widetilde X_H,\mathbb Z)$ and induced homomorphism$H^n(X,\mathbb Z) \rightarrow H^n(\widetilde X_H,\mathbb Z)$ for the coveringmap $p\colon \widetilde X_H \rightarrow X$ associated to a finite indexsubgroup $H < \pi_1X$, as well as the corresponding homology homomorphism.  We illustrate an open-source implementation of the algorithm by using it toshow that: (i) the degree $2$ homology group $H_2(\widetilde X_H,\mathbb Z)$distinguishes between the homotopy types of the complements $X\subset \mathbbR^4$ of the spun Hopf link and Satoh's tube map of the welded Hopf link (thesetwo complements having isomorphic fundamental groups and integral homology);(ii) the degree $1$ homology homomorphism $H_1(p^{-1}(B),\mathbb Z) \rightarrowH_1(\widetilde X_H,\mathbb Z)$ distinguishes between the homeomorphism types ofthe 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 havingisomorphic fundamental groups and integral homology).

Cohomology with local coefficients and knotted manifolds