We propose a two-dimensional carbon allotrope (named KT-graphene) by incorporating kagome and tetragonal lattices consisting of trigonal, quadrilateral, octagonal, and dodecagonal rings. The introduction of non-hexagonal rings can give rise to the localized electronic states that improve the chemical reactivity toward potassium, making KT-graphene a high-performance anode material for potassium-ion batteries. It shows a high theoretical capacity (892 mAh g-1), a low diffusion barrier (0.33 eV), and a low average open-circuit voltage (0.51 V). The presence of electrolyte solvents is propitious to boost the K-ion adsorption and diffusion capabilities. Moreover, one-dimensional nanotubes (KT-CNTs), rolled up by the KT-graphene sheet, are metallic regardless of the tube diameter. As the curvature increases, KT-CNTs exhibit significantly increased surface activity, which can promote the electron-donating ability of K. Furthermore, the curvature effect greatly enhances the efficiency of K diffusion on the inner surface compared to that on the outer surface.