The Gordian graph of knots has the knot isotopy classes as it's vertices, and an edge whenever you can pass from one knot to a other via a "finger move", equivalently if for some diagram of the knot you can perform a crossing change and pass to the other knot's isotopy class.
I've seen few theorems that strictly concern the Gordian graph's structure. Every vertex has infinite valence. Every vertex is contained in a complete graph of arbitrary size.
I have not seen a theorem that would contradict the graph being homogeneous -- having a graph automorphism sending any vertex to any other.
One could be even more specific, consider the graph to be a metric graph with edges of length one. I know of no theorems that contradict the assertion that for any two vertices of distance k, the spheres of radius p and q about the two points have infinite intersection as well as infinitely many points not in common (for all $k>0$) and $p+q\geq k$ with p and q positive. More generally if $n$ points have pairwise distance less than $r$, then any set from the Venn diagram of possible intersections (of the balls of radius $r$) is infinite.
As Misha suggests, is the graph hyperbolic?
Comment on the above: I think a succinct way to see why one would expect the Gordian graph to be non-hyperbolic is that satellite operations produce various maps of the gordian graph. Connect sum (with a fixed knot) produces an endomorphism of the gordian graph in the strict graph theoretic sense: vertices are sent to vertices, edges to edges. More elaborate satellite operations like Whitehead doubling produce cellular maps of the graphs and these would appear to be "expansive" maps with stretch factors, so that would rule out (one imagines) hyperbolicity. It's at least clear where vertices are sent, but the edge maps depend on a choice of decomposing "compound" finger moves. It's perhaps better to think of these as simply maps of the vertex sets, as metric spaces.
Is anyone aware of any work that would contradict any of these assertions?