Regular-equivalence of $2$-knot diagrams and sphere eversions

TL;DR

This paper introduces a method to construct pairs of 2-knot diagrams that require branch points in any Roseman move sequence, linking sphere eversions to diagram equivalence in 4-space.

Contribution

It provides a novel construction demonstrating that certain 2-knot diagram transformations inherently involve branch points, connecting sphere eversions to 4-dimensional isotopy constraints.

Findings

No sphere eversion can be lifted to a 4-space isotopy.

Constructed diagram pairs require branch points in any Roseman move sequence.

Establishes a new link between sphere eversions and 2-knot diagram equivalence.

Abstract

For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D'$ of the same knot such that any sequence of Roseman moves between $D$ and $D'$ necessarily involves branch points. The proof is done by developing the observation that no sphere eversion can be lifted to an isotopy in $4$-space.