The novel coronavirus SARS-CoV-2 infects human cells using a mechanism that involves binding and structural rearrangement of its spike protein. Understanding protein rearrangement and identifying specific residues where mutations affect protein rearrangement has attracted a lot of attention for drug development. We use a mathematical method introduced in (Baldwin2021) to associate a local topological/geometrical free energy along the SARS-CoV-2 spike protein backbone. Our results show that the total local topological free energy of the SARS-CoV-2 spike protein monotonically decreases from pre-to post-fusion and that its distribution along the protein domains is related to their activity in protein rearrangement. By using density functional theory (DFT) calculations with inclusion of solvent effects, we show that high local topological free energy conformations are unstable compared to those of low topological free energy. By comparing to experimental data, we find that the high local topological free energy conformations in the spike protein are associated with mutations which have the largest experimentally observed effect to protein rearrangement.