Newly replicated circular chromosomes are topologically linked. Controlling these topological changes, and returning the chromosomes to an unlinked monomeric state is essential to cell survival. XerCD-dif-FtsK recombination acts in the replication termination region of the Escherichia coli chromosome to remove links introduced during replication. We use topological methods to show definitively that there is a unique shortest pathway of unlinking by XerCD-dif-FtsK that strictly reduces the complexity of the links at every step. We delineate the mechanism of action of the enzymes at each step along this pathway and provide a 3D interpretation of the results.