On computing the first higher-order alexander modules of knots

Peter D. Horn

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

Cochran defined the nth-order integral Alexander module of a knot in the three-sphere as the first homology group of the knots (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. © 2014

Original languageEnglish (US)
Pages (from-to)153-169
Number of pages17
JournalExperimental Mathematics
Volume23
Issue number2
DOIs
StatePublished - Apr 3 2014

Keywords

  • 57M25
  • 57M27

ASJC Scopus subject areas

  • Mathematics(all)

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