  We systematically construct two-dimensional $\mathbb{Z}_3$ symmetry-protected
topological (SPT) three-state Potts paramagnets with gapless edge modes on a
triangular lattice. First, we study microscopic lattice models for the gapless
edge and, using the density-matrix renormalization group (DMRG) approach,
investigate the finite size scaling of the low-lying excitation spectrum and
the entanglement entropy. Based on the obtained results, we identify the
universality class of the critical edge, namely the corresponding conformal
field theory and the central charge. Finally, we discuss the inherent
symmetries of the edge models and the emergent winding symmetry distinguishing
between two SPT phases. As a result, the two topologically nontrivial and the
trivial phases define a general one-dimensional chain supporting a
tricriticality, which we argue supports a gapless SPT order in one dimension.
