A transverse knot invariant from Z2-equivariant Heegaard Floer cohomology

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ along based knots, and then prove that our transverse link invariant $c_{\mathbb{Z}_{2}}(\xi_{K})$ is an well-defined element which is always nonvanishing and functorial under certain classes of symplectic cobordisms, and describe its behavior under negative stabilization. It follows that we can use properties of $c_{\mathbb{Z}_{2}}(\xi_{K})$ to give a condition on transverse knots K which implies the vanishing/nonvanishing of the contact class $c(\xi_{K})$.