On Computing the First Higher-Order Alexander Modules of Knots

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)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.

Keywords::

• 57M25
• 57M27
• knot
• Alexander polynomial
• genus
• mutation

Notes

¹Available at http://www.math.uic.edu/culler/gridlink.

²A computer program for computing δ₁ is available at https://pdhorn.expressions.syr.edu/software/.

³See http://pdhorn.expressions.syr.edu/.

⁴A Sage worksheet for computing δ₁ is available at https://pdhorn.expressions.syr.edu/software/.