Rank 1 abelian normal subgroups of 2-knot groups

TL;DR

This paper investigates the structure of 2-knot groups with abelian normal subgroups of rank at least one, revealing conditions under which such groups lack minimal Seifert hypersurfaces or are equivalent to a known example.

Contribution

It characterizes 2-knot groups with certain abelian normal subgroups, linking algebraic properties to topological features and specific known examples.

Findings

If the group has a non-finitely generated abelian normal subgroup of rank ≥1, then either no minimal Seifert hypersurface exists or the group is topologically equivalent to a specific known example.

The paper establishes a connection between algebraic subgroup properties and the topological classification of 2-knots.

Abstract

If the group of a 2-knot group $K$ has an abelian normal subgroup of rank $\geq1$ which is not finitely generated then either $K$ has no minimal Seifert hypersurface or $K$ is topologically equivalent to Example 10 of Ralph Fox's``{\it A quick trip through knot theory}".