Using \({\mathbb {Z}}_3\) symmetry, we establish a topological condition for the existence of \({\mathbb {Z}}/2\) -harmonic 1-forms on Riemannian manifolds. As a corollary, if L is an oriented link in \(S^3\) with determinant zero, then there exists a non-degenerate \({\mathbb {Z}}/2\) -harmonic 1-form on the 3-fold cyclic branched covering of L. Furthermore, we find that there are infinitely many rational homology 3-spheres that admit a non-degenerate \({\mathbb {Z}}/2\) -harmonic 1-form.
