Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$.
For a fixed knot $K$ we can define the map $\varphi_K : X \rightarrow X$ as $\varphi_K (K^\prime) = K^\prime \sharp K$ .
It is easy to show that $ \forall K \in X$ we have $ d(K_1, K_2) \ge d(\varphi_K (K_1) , \varphi_K (K_2))$.
My question is: does the equality hold? Or in other words is the map $\varphi_K $ and isometry on $X$ with respect to $d$?