The methodology developed in this work is extensible to a number of applications that underpin fundamental modeling studies as well as theoretical/algorithm development. Prediction of the high-dimensional PEL topology can facilitate the development of coarse-grained potential energy functions that retain specific topological features at given energy thresholds for a variety of systems, such as polymers. This approach may also support new sampling approaches for complex energy landscapes. For example, identifying minima on complex landscapes that represent new polymorphs, isomers, or conformers of molecules and materials; alternatively, the accelerated observation of multiple time-scale dynamics in the absence of dimensionality reduction (for example using predicted topologies for learning methods). Finally, there are important relationships between energy landscape topology and the condition dependent population of states that contribute to the partition functions associated with macroscopic observables and thermodynamic behavior of chemical systems. In Section II we provide an introduction to sublevelset persistent homology in the context of intramolecular potential energy landscapes arising in chemistry. The data sets employed and mathematical methods are described in Section III. The main results are presented Section IV, followed by an application to more complex polymers in Section V, and concluding remarks in Section VI. Some of the mathematical proofs appear in Appendix A.