Subknots in ideal knots, random knots, and knotted proteins.

We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protocol. We characterize the sets of observed subknot types in global knots taking energy-minimized shapes such as KnotPlot configurations and ideal geometric configurations. We compare the sets of observed subknots to knot types obtained by changing crossings in the classical prime knot diagrams. Building upon this analysis, we study the sets of subknots in random configurations of corresponding knot types. In many of the knot types we analyzed, the sets of subknots from the ideal geometric configurations are found in each of the hundreds of random configurations of the same global knot type. We also compare the sets of subknots observed in open protein knots with the subknots observed in the ideal configurations of the corresponding knot type. This comparison enables us to explain the specific dispositions of subknots in the analyzed protein knots.