Seifert's algorithm, Châtelet bases and the Alexander ideals of classical knots

Abstract

I begin by developing a procedure for the construction of a Seifert surface, using Seifert's algorithm, and the calculation of a Seifert matrix for a knot from a suitable encoding of a knot diagram. This procedure deals with the inherent indeterminacy of the diagram encoding and is fully implementable. From a Seifert matrix one can form a presentation matrix for the Alexander module of a knot and calculate generators for the Alexander ideals. But to use the Alexander ideals to their full potential to distinguish pairs of knots one needs a Gröbner basis type theory for A = Z[t,t(-1)], the ring of Laurent polynomials with integer coefficients. I prove the existence of what I call Châtelet bases for ideals in A. These are types of Gröbner bases. I then develop an algorithm for the calculation of a Châtelet basis of an ideal from any set of generators for that ideal. This is closely related to Buchberger's algorithm for Gröbner bases in other polynomial rings. Using these algorithms and the knot diagram tables in the program Knotscape I calculate Châtelet bases for the Alexander ideals of all prime knots of up to 14 crossings. We determine the number of distinct ideals that occur and find examples of pairs of mutant knots distinguished by the higher Alexander ideals but not by any of the polynomials of Alexander, Jones, Kauffman or HOMELY.