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

Abstract

For each diagram $D$ of a $2$-knot, we provide a way to construct a new diagram $D^{\prime }$ of the same knot such that any sequence of Roseman moves between $D$ and $D^{\prime }$ 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.

10 pages, 6 figures

References in corpus (5)

• The compression theorem I
• Sphere eversions and realization of mappings
• Independence of Roseman moves including triple points
• The `Holiverse': holistic eversion of the 2-sphere in ${\mathbb{R}}^3$
• Knotted surfaces and equivalencies of their diagrams without triple points

Cited by in corpus (2)

• Independence of Roseman moves including triple points
• On rack colorings for surface-knot diagrams without branch points