2013/03/06 by Peter D. Horn, Horn, Peter D.
Computer Science · Mathematics · #57M25 #57M27 #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #msc:57M25 #msc:57M27 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1303.1545
20 pages, 3 figures, added some references, added result on mutation
openalex publication_date 2013/03/06 · arxiv created 2013/08/19 · arxiv updated 2013/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.