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](http://katlas.math.toronto.edu/wiki/File%3aTriquetra-Vesica-solid.png) has a lower simplicity
than [trefoil-2](http://katlas.math.toronto.edu/w/images/f/f2/3_1_AP.gif) (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? 



Is this knot invariant already treated somewhere in the literature?
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](http://dx.doi.org/10.1016/j.topol.2011.11.019)


