Moves on special spines

The main theorem of the theory of special spines states that any two spines of the same 3-manifold can be related be a sequence of moves $T^{\pm 1}$, where $T$ is performed in a regular neighborhood of an edge of a spine and increases the number of its true vertices by one. However, even in simple cases the proof of the theorem is not very helpful for finding explicit sequences of moves. We describe here a first nontrivial example of such sequence. The sequence relates two special spines of the 3-sphere with four removed 3-balls. This result answers a question of Scott Carter, who wanted to get such sequence.