Does the Gordian distance depend on the chosen link diagram?

Question

Let $K,L:S^1\rightarrow S^3$ 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?
Thanks in advance!