Bicontinuous cubic liquid crystalline (LC) phases are of particular interest due their possible applications in electronic devices and special supramolecular chirality. Herein, we report the design and synthesis of first examples of achiral bent-shaped polycatenar dimers, capable of displaying mirror symmetry breaking in their cubic and isotropic liquid phases. The molecules have a taper-shaped 3,4,5-trialkoxybenzoate segment connected to rod-like building unit terminated with one terminal flexible chain. The two segments were connected using an aliphatic spacer with seven methylene units to induce bending of the whole structure. Investigated by the small-angle X-ray scattering (SAXS), a double network achiral cubic phase Cub/Ia3 d, which is a meso-structure, and a chiral triple network cubic phase Cub/I23[*] are formed. The molecules self-assemble into molecular helices and progress along the networks. Interestingly, different linking groups such as ester or azo linkages and core fluorination lead to distinct local helicity, resulting in an alkyl chain volume dependent phase transition sequence Ia3d(L)-I23*-Ia3d(S). The re-entry of Iad phase and loss of supramolecular chirality is attributed to the delicate influence of steric effect at the mono-substitute end and interhelix interaction. Besides, aromatic core fluorination was proved to be a successful tool stabilizing the cubic phases in these dimers.