The ternary half-Heusler compounds have shown great potential for realizing new 3D topological insulators. With band gap tuning and spin orbit coupling these compounds may undergo topological phase transitions. In present work, we explore the possibility of realizing a topological insulating phase in half-Heusler family NaYO (Y = Ag, Au, and Cu). We find that for NaAgO, external strain ($\sim$19 \%) along with spin-orbit coupling, is required to achieve band-inversion at $\Gamma$ high-symmetry point and leads to phase transition from trivial to non-trivial topological insulating phase. In case of NaAuO and NaCuO, non-trivial phase appears in their equilibrium lattice constant, hence only spin-orbit coupling is enough to achieve band-inversion leading to non-trivial topology. The non-centrosymmetric nature of crystal geometry leads to the formation of two twofold degenerate point nodes near the Fermi level.

We study the electronic properties of coupled parallel Polyyne chains in a couple of symmetric stacking arrangements, namely the AA stacking and the AB stacking, with the single and triple carbon bonds of one chain aligned (AA) and anti-aligned (AB) with those of the other chain. Both these arrangements described by tight-binding Hamiltonians, whose parameters are calibrated by matching low energy dispersion provided by first principle calculations, fall in the BDI class of topological classification scheme. We calculate the topological invariants for all three topological phases of the system: one for the AA stacking and 2 for the AB one. In AA stacking, both the insulating and the metallic phase belongs to the same topological phase. Whereas, the model exhibits two different values of the topological invariant in the two different insulating phases (structurally differentiated by transverse strain). In this later stacking though the transition between two distinct topological phases with the closure of the gap is practically unachievable due to the requirement of the high transverse strain. We also show the existence of four non-zero energy edge modes in the AA stacking and that of two zero energy edge modes in one of the topological phases for the AB stacking.