Classifying the surface-knot modules

TL;DR

This paper characterizes the reduced first module of surface-knot modules of any genus using module properties and provides examples of fundamental groups that distinguish different genus surface-knots.

Contribution

It introduces a characterization of the reduced first module for surface-knots and constructs examples of fundamental groups unique to each genus.

Findings

Reduced first module characterized by finitely generated module properties

Explicit examples of fundamental groups for each genus g>0

Determination of torsion and torsion-free parts of the second module

Abstract

The $k$th module of a surface-knot of a genus $g$ in the 4-sphere is the $k$th integral homology module of the infinite cyclic covering of the surface-knot complement. The reduced first module is the quotient module of the first module by the finite sub-module defining the torsion linking. It is shown that the reduced first module for every genus $g$ is characterized in terms of properties of a finitely generated module. As a by-product, a concrete example of the fundamental group of a surface-knot of genus $g$ which is not the fundamental group of any surface-knot of genus $g-1$ is given for every $g>0$. The torsion part and the torsion-free part of the second module are determined by the reduced first module and the genus-class on the reduced first module. The third module vanishes. The concept of an exact leaf of a surface-knot is introduced, whose linking is an orthogonal sum of the torsion linking and a hyperbolic linking.