When topology becomes rare, standard Markov chain Monte-Carlo sampling is no longer effective because it simply never samples the rare topology-containing configurations. Starting from a standard code package for the implementation of lattice Quantum Chromodynamics, we developed two approaches to overcoming this problem. In one approach, we modify the Monte-Carlo weighting to enhance topological and “pretopological” configurations, enhancing the sampling (multiple topologies at a fixed temperature). The key advance in the first funding period was a method for better identifying the configurations which lie between non-topological and fully topological fields within lattice Quantum Chromodynamics, which allowed us to reweight so as to prefer these configurations and enhance the rate that the Markov chain moves between topologies. In the other approach, one compares topologies at one temperature, and then compares across temperatures separately within configurations of a fixed topology. By exploring how differently topological and non-topological configurations react to varying temperature, one can then determine the temperature dependence of topology (multiple temperatures at fixed topology).