Every regular knot projection admits a finite rational planar code

A smooth embedded circle has empty interior in $S^3$: in every submanifold chart it is a coordinate line, which contains no open three-dimensional ball. Choose $p\in S^3\setminus K$. Choose an orientation-preserving rotation $R\in SO(4)$ carrying $p$ to the fixed point at infinity; a rotation path from the identity to $R$ restricts to an ambient isotopy of $S^3$. Thus $R(K)$ misses infinity, and stereographic coordinates regard it as a compact smooth oriented knot in ${\mathbb{R}}^3$. Apply [F1] to this representative and list the finitely many crossings of the resulting projection. Around each crossing choose a closed disk meeting the image in exactly two transverse arcs, with all such disks pairwise disjoint. Subdivide the remaining compact arcs into finitely many subarcs contained in planar coordinate disks where each arc is a graph over a line interval and where nonincident pieces are separated. Compactness and finiteness give a positive minimum for the separations between the disjoint pieces.

In each crossing disk, choose a nearby rational point $q$ and move the crossing there by a small planar isotopy supported in that disk. Choose two linearly independent rational direction vectors close to the two tangent directions there. Small rational multiples in both signs give four rational endpoints about $q$, in the same cyclic order as the branch germs. Replace the branches near $q$ by the two straight diameters through these endpoints, joining back inside the corresponding four disjoint sectors; this is a small isotopy of the crossing. Outside these disks, approximate each graph arc from step 1.1 by a sufficiently fine polygonal graph and perturb its finitely many vertices to rational coordinates. Choose all perturbations smaller than one third of the positive separation bound, so distinct nonincident pieces remain disjoint and every adjacent segment stays in its graph chart.

Orient the polygonal edges by the original circle parameter, pair the four crossing half-edges as on the original branches, and retain each crossing's over/under label. In each graph chart interpolate linearly between the original graph and the rational polygonal graph; the graphs remain embedded, their boundary endpoints agree, and their disjoint neighborhoods prevent new intersections. In each crossing disk the two interpolated branches keep the same four sectors and nonzero crossing determinant at the fixed rational point. Smooth each bend inside a smaller disk disjoint from all crossings. This gives an isotopy through diagrams with exactly the original crossing data. Lift the over/under heights continuously; strict height inequalities at each crossing persist, so this is an isotopy of embedded circles, which extends to an ambient isotopy by [F4]. The resulting rational polygonal data satisfy all clauses of [F2].