The $\mathbb{ Z}_{ 2}$-equivariant Heegaard Floer cohomlogy $\widehat{ HF}_{ \mathbb{ Z}_{ 2}}(\Sigma(K))$ 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 $\widehat{ HF}_{ \mathbb{ Z}_{ 2}}(\Sigma(K))$ 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 $\hat{ \mathcal{ T}}_{ \mathbb{ Z}_{ 2}}(K)$ as an element of $\widehat{ HFK}_{ \mathbb{ Z}_{ 2}}(\Sigma(K),K)$, which is a refinement of $\widehat{ HF}_{ \mathbb{ Z}_{ 2}}(\Sigma(K))$, and show that it is a refinement of both the LOSS invariant $\hat{ \mathcal{ T}}(K)$ and the $\mathbb{ Z}_{ 2}$-equivariant contact class $c_{ \mathbb{ Z}_{ 2}}(\xi_{ K})$.
Categories: math.GT