In our investigation, we considered grid diagrams with complexity between 5 (the minimal GN in which nontrivial knots appear) and 20. We created the network of these grid diagrams where two grids of the same GN are connected by a directed edge if it is possible to transform the first diagram into the second via a single strand passage (i.e., an interleaving commutation). For each GN, the network of grid diagrams has finitely many vertices and edges. The strand passage–mediated flux (or knot interconversion flux) from a knot type to another is the union of directed edges in the network going from the subspace corresponding to the first knot type into the subspace of the second one. The intensity of the flux is proportional to the number of these directed interconversion edges. We performed our analysis by enumerating knot conformations and by counting unbiased and hooked strand passages between conformations of given knot types. Imposing that the strand passages happen only at hooked juxtapositions changes the shape of the network of configurations. The subspaces corresponding to simple knot types become more preferable than the others, as we discussed in Results. Since the GN correlates with the length of the underlying closed polymer, the configuration space changes as GN increases in value (see Fig. 5), which is similar to models that take length as a parameter (27). (We remark that our model, being purely topological, does not consider such physical quantities such as temperature. In other words, the model is temperature independent, and each conformation is assigned the same statistical weight.)

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