Knotted surfaces and equivalencies of their diagrams without triple points

TL;DR

This paper investigates the equivalence of knotted surfaces in four-dimensional space via their projections without triple points, introducing an invariant that shows not all such diagrams are equivalent.

Contribution

It introduces an invariant for p-diagrams of surface links and provides a counterexample demonstrating that not all equivalent diagrams are p-equivalent.

Findings

An invariant distinguishes p-diagrams of surface links.

Counterexample shows some equivalent surface links are not p-equivalent.

The study clarifies limitations of diagrammatic moves without triple points.

Abstract

The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent surface links are called p-equivalent if there exists a finite sequence of local moves, such that each of them is one of the four moves taken from the seven on the well known Roseman list, that connects only p-diagrams. It is natural to ask: are any of two p-diagrams of equivalent surface links always p-equivalent? We introduce an invariant of p-equivalent diagrams and an example of linked surfaces that answers our question negatively.