The tyrosine family of recombinases produces two smaller DNA circles when acting on circular DNA harboring two recombination sites in head-to-tail orientation. If the substrate is supercoiled, these circles can be unlinked or form multiply linked catenanes. The topological complexity of the products varies strongly even for similar recombination systems. This dependence has been solved here. Our computer simulation of the synapsis showed that the bend angles, phi, created in isolated recombination sites by protein binding before assembly of the full complex, determine the product topology. To verify the validity of this theoretical finding we measured the values of phi for Cre/loxP and Flp/FRT systems. The measurement was based on cyclization of the protein-bound short DNA fragments in solution. Despite the striking similarity of the synapses for these recombinases, action of Cre on head-to-tail target sites produces mainly unlinked circles, while that of Flp yields multiply linked catenanes. In full agreement with theoretical expectations we found that the values of phi for these systems are very different, close to 35 degrees and 80 degrees, respectively. Our findings have general implications in how small protein machines acting locally on large DNA molecules exploit statistical properties of their substrates to bring about directed global changes in topology.