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?

5more comments