SYMMETRIES OF MINIMUM GRAPHS IN THE 3-SPHERE

Abstract

In this paper, a new equivalence relation, weak isotopy, on spatial graphs will be introduced. By using a rearrangement theorem (Theorem 1), it is shown that, for any spatial embedding Γ of a connected graph, the weak isotopy class [Γ] w.i. contains a unique minimum element Γ min up to ambient isotopy. We will see that this minimum element plays an important role in investigating symmetries of embeddings in [Γ] w.i. .