Surfaces and their Profile Curves

TL;DR

The paper analyzes how the knottedness of an embedded surface $F$ in $\mathbb{R}^3$ relates to the knotting of its fold curves under projection to $\mathbb{R}^2$. It proves two main results: (i) any embedded genus-$g$ surface can be isotoped so that its fold curves form an unlink with at most $g$ components, via fold-curve band sums using pockets and a basic sphere construction; and (ii) introduces the invariant $\delta(\gamma,F)=\lambda(\gamma,F)-w(\gamma)$, which provides a complete obstruction to turning a curve $\gamma$ on $F$ into a fold curve by an isotopy fixing $\gamma$, with $\delta=0$ exactly when such an isotopy exists. The work further shows how $\lambda(\cdot)$ and $w(\cdot)$ govern realizability of curves as fold curves, yielding both realizability results for knots on unknotted surfaces and geometric obstructions for certain curves on spheres or knotted tori. These insights advance understanding of projection singularities, surface topology, and 3-D reconstruction from silhouettes.

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.

Figures (11)

• Figure 1: Fold curves project to the edges of a 2-dimensional image.
• Figure 2: A projection of a generic surface has singularities along a collection of fold curves. These are smooth regular curves whose projection to the plane gives a collection of piecewise-smooth curves, the apparent contour.
• Figure 3: A knotted and unknotted torus with the same pair of fold curves, shown at right.
• Figure 4: A knotted torus in ${\mathbb{R}}^3$ with two fold curves, each representing a trefoil knot.
• Figure 5: A fold-curve band sum

Theorems & Definitions (21)

• Theorem 1
• proof
• Lemma 2
• proof
• Corollary 3
• proof
• Corollary 4
• Lemma 5
• proof
• Theorem 6