Every embedded planar graph gives rise to a link (collection of knots) according to a particular procedure (I've heard it called the Mercat procedure). I am trying to determine whether this construction is universal. So far, I have constructed a family of alternative procedures for turning planar graphs into links; I am trying to show that the Mercat procedure is optimal among these alternatives. One hypothesis is that the Mercat procedure produces a link with the fewest number of components, but I am having trouble seeing whether this is true or gaining intuition about it. Details follow.
The "dart space" of an embedded undirected graph $G=(V,E)$ is the set consisting of two points for every edge $e\in E$; intuitively, it consists of the set of edges where one or the other of its endpoints has been marked as special. Categorically, we can identify the dart space of $G$ with the set of graph morphisms $\hom(\mathbf{2}, G)$, where $\mathbf{2}$ is the graph with two vertices and one edge.
The Mercat procedure uses the fact that every graph can be characterized by two permutations on its darts. Let $\alpha:D\rightarrow D$ be the involution which, for each dart, flips which endpoint is marked as special. Because the graph is embedded, let $r:D\rightarrow D$ be the permutation which sends each dart $\langle e, v\rangle$ to its immediate clockwise neighbor $\langle e^\prime, v\rangle$, turning around $v$. These data comprise a combinatorial map, and completely characterize the embedded graph.
We can describe the Mercat procedure as a permutation on the space $D \times \{\mathsf{over},\mathsf{under}\}$, which equips each dart with a pair of crossing threads, one over and one under. Equivalently, we can describe it as a permutation on $D+D$.
Let $\beta$ be the involution on $D+D$ which exchanges $(d,\mathsf{over})\leftrightarrow (d,\mathsf{under})$. Then the Mercat procedure is defined as $$\mathcal{M}:D+D\xrightarrow{r + r^{-1}}D+D \xrightarrow{\alpha + \alpha} D+D \xrightarrow{\beta} D+D$$ and the components of the associated link are the orbits of this procedure.
We can generalize this procedure, and hopefully show that the Mercat procedure is an optimal choice among this newly-created family of procedures, essentially by replacing the embedded rotation $r$ with any permutation that behaves in a pivot-like way, i.e. by mapping each dart to a dart with the same designated endpoint.
Categorically, we can formalize this condition by fixing an arrow $\mathsf{pivot}:\mathbf{1}\rightarrow \mathbf{2}$ which identifies which of the endpoints of a dart is the "pivot" endpoint. Then a pivot-like endomorphism $\rho$ of $D$ is just one where every dart has the same special endpoint as its image under $\rho$.
$$\hom(\mathbf{2},D) \xrightarrow{\rho \circ -} \hom(\mathbf{2},D) \xrightarrow{-\circ \mathsf{pivot}} \hom(\mathbf{1}, D) \quad=\quad \hom(\mathbf{2},D) \xrightarrow{-\circ \mathsf{pivot}} \hom(\mathbf{1}, D) $$
Such an endomorphism $\rho$ no longer needs to be spatially realizable or coherent, so we can abandon the requirement that $G$ be concretely embedded in a surface. For each pivot-like endomorphism $\rho:D\rightarrow D$, we have a corresponding Mercat-like procedure
$$D+D\xrightarrow{\rho + \rho^{-1}}D+D \xrightarrow{\alpha + \alpha} D+D \xrightarrow{\beta} D+D$$ and the components of the associated link are the orbits of this procedure.
The generality of $\rho$ allows for some bizarre and degenerate knotwork constructions. For example when $\rho = \mathsf{id}_D$, there is a pair of intersecting loops wrapped around every edge, forming $|E|$ separate components.
Within this context, I am wondering whether I can show that there is something optimal/special about the Mercat procedure, e.g. whether all choices of $\rho$ produce links with fewer components. Certainly the pivot $r$ seems like the least-degenerate sort of pivot permutation on darts, as it has no fixed points except for when obligatory—in cases like loops and vertices of degree 1. I wonder if I can show that all such minimally-degenerate pivot permutations produce links with the same number of components (e.g. using a deletion-contraction recurrence), even if some of these pivot permutations are not physically realizable as coherent rotations.