A new obstruction for ribbon-moves of 2-knots: 2-knots fibred by the punctured 3-tori and 2-knots bounded by homology spheres

TL;DR

This paper introduces a new obstruction criterion for ribbon-move equivalence of 2-knots, showing that certain fibered 2-knots cannot be equivalent to knots with different Seifert hypersurfaces, thus advancing understanding of 2-knot transformations.

Contribution

It provides a novel obstruction for ribbon-move equivalence based on the fibered structure and Seifert hypersurface properties of 2-knots.

Findings

Fibered 2-knots with punctured 3-tori as fibers are not ribbon-move equivalent to knots with punctured homology sphere Seifert hypersurfaces.

The main theorem establishes a new criterion distinguishing classes of 2-knots under ribbon-move equivalence.

Abstract

This paper gives a new obstruction for ribbon-move equivalence of 2-knots. Let $K$ and $K'$ be 2-knots. Let $K$ and $K'$ are ribbon-move equivalent. One corollary to our main theorem is as follows. A 2-dimensional fibered knot whose fiber is the punctured 3-dimensional torus is not ribbon-move equivalent to any 2-dimensional knot whose Seifert hypersurface is a punctured homology sphere.