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We present comprehensive theoretical studies on the lattice relaxation andthe electronic structures in general non-symemtric twisted trilayer graphenes.By using an effective continuum model, we show that the relaxed latticestructure forms a patchwork of moir\'e-of-moir\'e domains, where a moir\'epattern given by layer 1 and 2 and another pattern given by layer 2 and 3become locally commensurate. The atomic configuration inside the domainexhibits a distinct contrast between chiral and alternating stacks, which aredetermined by the relative signs of the two twist angles. In the chiral case,the electronic band calculation reveals a wide energy window ($>$ 50 meV) withlow density of states, featuring sparsely distributed highly one-dimensionalelectron bands. These one-dimensional states exhibit a sharp localization atthe boundaries between super-moir\'e domains, and they are identified as atopological boundary state between distinct Chern insulators. The alternatingtrilayer exhibits a coexistence of the flat bands and a monolayer-like Diraccone, and it is attributed to the formation of moir\'e-of-moir\'e domainsequivalent to the mirror-symmetric twisted trilayer graphene.

Multi-scale lattice relaxation in general twisted trilayer graphenes