Many physical systems exhibit complex spatio-temporal behaviors that are difficult to characterize. We describe an approach that uses topological tools (specifically, computational homology) to connect experimentally observed structures to underlying dynamics. As a specific example, we will discuss homological characterizations of spiral defect chaos, a weakly turbulent state of Rayleigh-Benard convection. We observe asymmetries between hot and cold flows and show novel measures of boundary influence and indicators of system control parameters. We also find the evolution of the global structure of the flow to be primarily stochastic unlike the locally chaotic signatures reported previously.