The structure and stability of nonlinear carbyne carbon clusters was estimated on the basis of ab initio quantum mechanical calculations. The catenane (catecarbynes), cyclic (cycarbynes), and knotted (knocarbynes) structures of closed polycarbon chains containing 60 carbon atoms are described. Comparison of the calculated C-13 NMR spectra with that of C-60 fullerene provides a basis for experimental identification of C-60 topological isomers in the mixture.