${\mathbb{Z}}_2$-equivariant Heegaard Floer cohomology of knots in $S^3$ as a strong Heegaard invariant

The ${\mathbb{Z}}_2$-equivariant Heegaard Floer cohomlogy ${\widehat{HF}}_{{\mathbb{Z}}_2}(\mathrm{\Sigma }(K))$ of a knot $K$ in $S^3$, constructed by Hendricks, Lipshitz, and Sarkar, is an isotopy invariant which is defined using bridge diagrams of $K$ drawn on a sphere. We prove that ${\widehat{HF}}_{{\mathbb{Z}}_2}(\mathrm{\Sigma }(K))$ can be computed from knot Heegaard diagrams of $K$ and show that it is a strong Heegaard invariant. As a topolocial application, we construct a transverse knot invariant ${\widehat{\mathcal{T}}}_{{\mathbb{Z}}_2}(K)$ as an element of ${\widehat{HFK}}_{{\mathbb{Z}}_2}(\mathrm{\Sigma }(K),K)$, which is a refinement of ${\widehat{HF}}_{{\mathbb{Z}}_2}(\mathrm{\Sigma }(K))$, and show that it is a refinement of both the LOSS invariant $\widehat{\mathcal{T}}(K)$ and the ${\mathbb{Z}}_2$-equivariant contact class $c_{{\mathbb{Z}}_2}({\xi }_K)$.