The configuration space available to randomly cyclized polymers is divided into subspaces accessible to individual knot types. A phantom chain utilized in numerical simulations of polymers can explore all subspaces, whereas a real closed chain forming a figure-of-eight knot, for example, is confined to a subspace corresponding to this knot type only. One can conceptually compare the assembly of configuration spaces of various knot types to a complex foam where individual cells delimit the configuration space available to a given knot type. Neighboring cells in the foam harbor knots that can be converted into each other by just one intersegmental passage. Such a segment-segment passage occurring at the level of knotted configurations corresponds to a passage through the interface between neighboring cells in the foamy knot space. Using a DNA topoisomerase-inspired simulation approach we characterize here the effective interface area between neighboring knot spaces as well as the surface-to-volume ratio of individual knot spaces. These results provide a reference system required for better understanding mechanisms of action of various DNA topoisomerases.