Surfaces and their Profile Curves

TL;DR

This paper explores the relationship between embedded surfaces in three-dimensional space and their fold curves, showing how surfaces can be isotoped to simplify fold curves and introducing a new invariant to analyze curve transformations.

Contribution

It introduces a method to isotope any knotted surface so its fold curves become an unlink and defines a new invariant to obstruct certain fold curve transformations.

Findings

Surfaces can be isotoped to have unlinked fold curves.

A new invariant provides a complete obstruction to transforming fixed curves into fold curves.

The results connect surface knotting with curve invariants.

Abstract

This paper examines the relationship between the knotting of an embedded surface in $\R^3$ and the knotting of its fold curves, formed by the singular set of projection to a plane. The first result shows that every surface, no matter how knotted, can be isotoped so that its fold curves form an unlink. A second result defines a new invariant which gives a complete obstruction to turning a fixed curve on a surface into a fold curve.