In nuclear science, density functional theory (DFT) is a powerful tool to model the complex interactions within the atomic nucleus, and is the primary theoretical approach used by physicists seeking a better understanding of fission. However DFT simulations result in complex multivariate datasets in which it is difficult to locate the crucial `scission' point at which one nucleus fragments into two, and to identify the precursors to scission. The Joint Contour Net (JCN) has recently been proposed as a new data structure for the topological analysis of multivariate scalar fields, analogous to the contour tree for univariate fields. This paper reports the analysis of DFT simulations using the JCN, the first application of the JCN technique to real data. It makes three contributions to visualization: (i) a set of practical methods for visualizing the JCN, (ii) new insight into the detection of nuclear scission, and (iii) an analysis of aesthetic criteria to drive further work on representing the JCN.