On computing higher-order Alexander modules of knots

Cochran defined the \(n\)th-order integral Alexander module of a knot in the three sphere as the first homology group of the knot’s (n+1)th-iterated abelian cover. The case \(n=0\) gives the classical Alexander module (and polynomial). After a localization, one can get a finitely presented module over a principal ideal domain, from which one can extract a higher-order Alexander polynomial. We present an algorithm to compute the first-order Alexander module for any knot. Included in this algorithm is a solution to the word problem in finitely presented \(\mathbb{Z}[\mathbb{Z}]\)-modules.

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