TITLE: Is this knot invariant already treated somewhere in the literature?
QUESTION [8 upvotes]: Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K  \} /  \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by Reidemeister moves: 
$d(D_0,D_1) = $ minimum length of a sequence of Reidemeister moves (each move is possibly followed by a planar isotopy), and $D_0,D_1$ are two diagrams
 of $K$.
We can build a graph $\mathcal{G}_K$ as follows: the vertices are given by the points of $Y_K$, while the 
edges are given by the pairs $(D_0,D_1)$ such that $d(D_0,D_1)=1$. 
This construction parallels Hirasawa and Yoshiaki's in  "The Gordian Complex of Knots" for
the Gordian metric space of  knots, but unlike the Gordian complex, each vertex of $\mathcal{G}_K$ 
 has only finitely many edges (the number of edges at a vertex $D$ corresponds  to the possible 
inequivalent diagrams - up to planar isotopy - that can be reached from $D$ by a single 
Reidemeister move).
For each knot $K$ we can thus define a function $$\varphi_K : Y_K \longrightarrow \mathbb{N}$$ 
called the simplicity, as $\varphi_K (D) = \#\{$edges of $\mathcal{G}_K$ that contain $D$ in their boundary$\}$.
As an example the "standard" diagram of a trefoil trefoil-1 has a lower simplicity
than trefoil-2 (you need to take into account that the former is 3-symmetric).
Taking the minimum of the simplicity over all diagrams of a knot yields an invariant $\mathcal{S}(K)$.
Is the invariant $\mathcal{S}(K)$, or any possible application/connection, known?

REPLY [2 votes]: I think the minimal degree of "Reidemeister complex" has some meaning.
I know one paper about this topic.
A distance for diagrams of a knot