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We consider a real-valued continuous function on the 3-sphere cross time. At every moment, the function defines a Reeb graph which is free of loops because the 3-sphere is simply connected. The evolution through time defines a continuous series of Reeb graphs. In this work we classify the combinatorial changes that modify the Reeb graph when the critical points of the function undergo non-generic configurations. We also develop an algorithm that constructs the series of Reeb graphs, representing it as a persistent data structure. The most important remaining question is how to simplify the series as well as the Reeb graphs in it. Equivalently, which is the most appropriate generalization of the concept of topological persistence to time-varying data?