Classifying the surface-knot modules

Abstract

The kth module of a surface-knot of a genus g in the 4-sphere is the kth 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.

The authors' abstract, as published at the source. Journal of Knot Theory and Its Ramifications, 2026 · DOI ↗