On Computing the First Higher-Order Alexander Modules of Knots

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Cochran defined the nth-order integral Alexander module of a knot in the three-sphere as the first homology group of the knot’s (n + 1)st iterated abelian cover. The case n = 0 gives the classical Alexander module (and polynomial). After a localization, one can obtain 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. As applications, we show that these higher-order Alexander polynomials provide a better bound on the knot genus than does the classical Alexander polynomial, and that they distinguish mutant knots. Included in this algorithm is a solution to the word problem in finitely presented -modules.