Several assembly theorems, for obtaining the linkage characteristic polynomial for a complex chain through a series of steps involving the known polynomials for subunits of the chain, are derived and presented. These theorems give insight into how the topological information concerning the linkage is stored in the polynomial and might contribute to the automated recognition of linkage structure in generalized computer-aided design programs. Based on graph theory, the characteristic polynomial cannot characterize the graph up to isomorphism. However, for practical applications in the field of linkage mechanisms, it is extremely likely that the characteristic polynomials are unique for closed connected kinematic chains without any overconstrained subchains.

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