In recent years, it has become possible via on-surface bottom-up synthesis to engineer the topological character of carbon nanostructures. Graphene nanoribbons and 1D conjugated polymers (1DCPs) have thus tailored so as to host either topologically trivial or non-trivial phases. Molecular design is the primary means to set the topological class of these nanomaterials. However, external control over topology is also demonstrated via electric fields or top-down hydrogenation. Inspired by the connection between topology and π-conjugation, here it is demonstrated via first-principles calculations that aryl ring twist angles also serve as topological knobs. Focusing on rationally designed 1DCPs composed of triarylmethyl (TAM) units, it is shown that rotation of certain aryl rings enables a transition from the trivial to the topologically non-trivial phase. Accordingly, fixing a particular twist angle configuration (e.g., via chemical functionalization) is equivalent to robustly setting a targeted topological phase. It is also found that in considered 1DCPs, the quantum phase transition occurs without the electronic band gap closing, due to a multiradical antiferromagnetic phase emerging at the transition point. All in all, this study highlights the potential of aryl ring twisting for engineering topological properties in carbon nanomaterials and establishes TAM 1DCPs as exotic topological 1D systems.