SYMMETRIES OF MINIMUM GRAPHS IN THE 3-SPHERE

In this paper, a new equivalence relation, weak isotopy, on spatial gr aphs will be introduced. By using a rearrangement theorem (Theorem 1), it is shown that, for any spatial embedding Gamma of a connected grap h, the weak isotopy class [Gamma](w.i.) contains a unique minimum elem ent Gamma(min) up to ambient isotopy. We will see that this minimum el ement plays an important role in investigating symmetries of embedding s in [Gamma](w.i.).