Linear-size good exterior triangulation from a planar knot code

If $n>0$, there are $4n$ darts and $2n$ paired edges. The connected crossing graph has $V=n$, $E=2n$, and hence $F=2-V+E=n+2$ faces by [F2]. The sum of face-boundary port occurrences, counted with multiplicity, is $2E=4n$. Choose disjoint small disks around the graph vertices and disjoint narrow rectangular bands along its edges, attached in the prescribed cyclic order. Their union $N(G)$ is a connected planar subsurface of the sphere; each boundary component is one face cycle, and the closure of each complementary component is a disk. The total number of vertex-disk ports, band sides and boundary occurrences is bounded by a universal constant times $n$.

Triangulate each vertex disk and edge band by fixed templates. Subdivide every side occurrence on each complementary face boundary circle by a fixed number of new vertices and cone the resulting simple polygon to one interior vertex. Each face of boundary length $d$ contributes at most $c_{0}d$ triangles for a fixed $c_0$, and $\sum d=O(n)$ by step 1.1. Since the face boundary circles of $N(G)$ are embedded circles with port occurrences kept distinct, these are regular disk triangulations; matching subdivisions along shared ports and one fixed barycentric subdivision give a genuine finite simplicial triangulation of the projection sphere with $O(n)$ triangles.

Take the product of this triangulated sphere with an interval. Each triangular prism is split into three tetrahedra by the fixed staircase subdivision; cap the two ends by coning their triangulated boundary spheres to obtain a triangulated $S^3$ with $O(n)$ tetrahedra. In each crossing disk, place the over- and under-strand arcs at distinct interval heights and connect each pair of opposite ports; along each edge band connect its end ports by a linear arc whose height interpolates between the endpoint heights. In a crossing block the two arcs are disjoint because their heights differ; interiors of distinct edge blocks are disjoint, and every edge arc joins only its designated crossing ports. A fixed subdivision of each crossing and edge block makes these finitely many local arc templates subcomplexes. For the supplied $P$, choose the disks and bands along its actual projection. Carry the templates to its actual strand heights by increasing piecewise-linear maps of the interval fibres, fixed at the top and bottom; at a crossing the prescribed two heights have the same order, so both can be matched. Choose the maps identically on shared port collars and extend over the remaining face regions, triangulating the finitely many height graphs and interpolating between successive levels. Extend over the two end balls by coning the boundary maps. These maps give a PL homeomorphism of the template sphere carrying its template knot to the supplied $P$. Transport the triangulation through it. Its abstract simplex count remains $O(n)$, even if the PL parametrizations require further subdivisions in ambient coordinates; thus the same $P$ is a subcomplex.

Around each vertex and edge of the 1-subcomplex $P$, take the fixed ball and tube blocks from the definition of a good exterior triangulation. Choose a normal 2-frame at one point, extend it across the finitely many edge and vertex blocks, and adjust the final frame along the last edge by a path of rotations in $SO(2)$ to agree with the initial frame. Thus their union is the product $S^1\times D^2$, a solid torus with triangulated boundary. The number of blocks and tetrahedra per block is constant, so removing its interior leaves a compact triangulated exterior with $O(n)$ tetrahedra. At an interior point of a chosen knot edge, the fixed local tube template has a meridian circle using a fixed number of boundary edges; mark that path.

If $n=0$, use the standard genus-one product decomposition of $S^3$ into two solid tori and the fixed product triangulation of the exterior solid torus as a triangle times a three-edge circle. Splitting its three triangular prisms into three tetrahedra each and marking the three-edge circle $\{a_0\}\times S^1$ on its boundary gives a fixed-size model: this is the original knot's meridian, meeting an exterior disk boundary once. For a supplied crossingless $P$, parametrize its polygonal projection disk and the complementary sphere disk and match its height graph by the same fibre construction as in step 3.1; transport the fixed model through the resulting PL homeomorphism. Increasing $C$ to dominate this model and all fixed local template constants in steps 2.1, 3.1 and 4.1 yields both stated $C(n+1)$ bounds. All code-dependent choices use the finite label and port order, so the construction is explicit.