Finite planar codes for oriented knot diagrams

Work with smooth oriented knots in the standard oriented three-sphere; a knot is a one-component oriented link. A finite planar code for an oriented knot diagram consists of finite vertex and edge sets, incidence data, a cyclic order of the four half-edges at every crossing vertex, a pairing of opposite half-edges into the two local branches, an orientation of each edge, and a choice of which branch is over at each crossing. Every other vertex is a bend vertex of valence two. The code includes a finite plane-drawing certificate: rational coordinates for the vertices and bends, and a finite polygonal path for every edge. The certificate must realize the incidences and cyclic orders, have no edge self-intersection, and have pairwise disjoint edge interiors except at the specified bend and crossing vertices; at a crossing the two branches must be transverse. These conditions are checked by finitely many exact segment-intersection tests. Edge orientations must join coherently across each paired branch and each bend vertex, and following the edges through the crossing pairings and the degree-two bend vertices must give one oriented circle. Thus an admissible code represents a connected regular oriented diagram of a single knot. If there are no crossing vertices, the code is one oriented polygonal circle and has one arc. Smooth each bend in a small disk disjoint from all crossings and other edges; the resulting smooth diagram has the same crossing data.

An under-arc is a maximal portion of the oriented circle between consecutive points where the knot passes under at a crossing; the over branch is not cut. With no crossings the entire circle is the unique under-arc. A positive meridian of an oriented branch is the oriented boundary of a small normal disk, with disk orientation chosen so that the oriented tangent to the branch followed by the disk orientation agrees with the ambient orientation of $S^3$. Planarity, incidence, orientation consistency, one-component tracing and the over/under data are input validity conditions; a procedure receiving an invalid finite record must report invalid input rather than repair it. The code stores no algebraic crossing sign: the oriented plane and the direction of the over branch determine which incident under branch is on its left and which is on its right.

Smooth oriented links in the standard three-sphere and the one-component knot convention

Finite transverse double points, strand orientations, and over/under data in a regular oriented diagram