On computing higher-order Alexander modules of knots

Peter D. Horn

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arXiv · 1303.1545

On computing higher-order Alexander modules of knots

Peter D. Horn

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Abstract

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)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. 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 detect mutation. Included in this algorithm is a solution to the word problem in finitely presented Z[Z]-modules.

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BibTeX RIS

Peter D. Horn. 2013-08-19. On computing higher-order Alexander modules of knots. https://arxiv.org/abs/1303.1545

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