Regular oriented link diagrams

Let $\pi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^3\to {\mathbb{R}}^2$ be the orthogonal projection forgetting the last coordinate, $\pi (x_1,x_2,x_3):=(x_1,x_2)$, and let $L\text{\hspace{0.17em}⁣}:C\to S^3$ be an oriented link contained in ${\mathbb{R}}^3\subset S^3$ (Oriented links in the three-sphere and ambient isotopy, so $C$ is a finite disjoint union of oriented circles and $L$ is a smooth embedding).

The projection $\pi \circ L$ is regular when:

1. it is an immersion, that is, the projected derivative never vanishes (Immersions, submersions, and constant-rank maps);
2. its only multiple points are finitely many double points, and at every double point the two branches meet transversely: the two projected tangent lines are distinct (Transverse smooth maps);
3. there are no triple or higher multiple points: no point of the plane is the image of three or more distinct source points.

A regular oriented link diagram $D$ consists of the image $\pi (L)$ together with

(a) the over/under datum at every double point: which of the two branches lies above the other, read from the forgotten third coordinate (the branch with the larger $x_3$-coordinate at the crossing is drawn over); and (b) the orientations of the strands, induced by the orientation of $L$.

Thus a diagram is a decorated immersed oriented planar graph with $4$-valent crossing vertices, together with any components having no vertices. Its edges are the strands between consecutive crossings, and the decoration records which strand passes over at each vertex. A diagram is understood up to planar isotopy (Planar isotopy of link diagrams); two links that differ by an ambient isotopy moved off $\mathrm{\infty }$ may be represented by diagrams differing by planar isotopy and the local moves of Oriented Reidemeister moves, which is the content of the Reidemeister equivalence theorem below.

Finiteness is part of the definition. Condition (2) is the clause that makes the finitely many double points available for the slicing arguments used in the Reidemeister and Markov proofs; a projection with infinitely many double points, accumulations, or tangential double points is not regular and is not used below. The definition fixes the ambient plane, the projection and the over/under convention once and for all.