Let K,L:S¹ rightarrow S³ be two knots. The Gordian distance between K and L is defined to be the minimum number of crossing changes to convert a knot diagram of K to a knot diagram of L. Further I understand this definition as taking the minimum over all such knot diagrams of K. However the question arises whether or not the Gordian distance does depend on the chosen knot diagram. More precisely: Does the Gordian distance change under Reidemeister moves?