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Reconnection processes appear at widely different scales, from microscopic DNA recombination to large-scale reconnection of vortices in fluid turbulence. In Escherichia coli DNA replication yields two interlinked circular chromosomes, returning the chromosomes to an unlinked monomeric state is essential to cell survival. This process is typically mediated by topoIV, a typeII topoisomerase. In the absence of topoIV, site-specific recombinases XerC/D co-localize with translocase FtsK in the replication termination region to remove replication links. We recently provided mathematical proof that there is a unique minimal pathway of DNA unlinking by local reconnection assuming that at every step the topological complexity goes down. We here investigate whether there are other minimal pathways of unlinking replication links by local reconnection when we relax the complexity assumption. We first determine analytically that there are exactly nine minimal reconnection pathways for unlinking the 6-cat if we assume no increase in complexity at every step. Then we eliminate the assumption and embark on a numerical exploration of unlinking pathways. We introduce a Monte Carlo method to simulate local reconnection, provide a quantitative measure to distinguish among pathways and conclude that the unique unlinking pathway found under the strict assumption remains the most probable after the assumption is lifted. These results point to a universal property relevant to any local reconnection event between two sites along one or two circles, such as the reconnection of knotted fluid vortices.