remains true if the genus of the fiber is one, provided that we take coefficients in an algebra over $\Lambda_\omega$, where $\Lambda_\omega$ is the Novikov field viewed as a module over ${{\mathbb F}}[H^1(Z_0)]$ via a choice of closed 2-form $\omega$ which evaluates non-trivially on the capped-off fiber.

In the Novikov twisted case, the primary ground algebra for our purposes is $\Lambda_\omega[C_n]$, where $\omega$ is Poincar[é]{} dual to the class of the fiber. We refer to Section \[trianglesec\] for details regarding Novikov ring coefficients.

In [@Contact Proposition 3.1], [[Ozsváth-Szabó]{} ]{}construct a Heegaard triple diagram for the surgery cobordism $W: -Z_0(L)\to -Z$ with the following properties:

-   The diagram is weakly admissible for the unique [[ ]{}]{}structure on $W$ extending ${\mathfrak{s}}_{1-g}$.

-   There are precisely two intersection points ${\mbox{\bf u}}$, ${\mbox{\bf v}}$ providing generators for the chain complex $CF^+(-Z_0(L), {\mathfrak{s}}_{1-g})$, and the only nontrivial differential is $\partial^+(U^{-j}{\mbox{\bf u}}) = U^{-j+1}\cdot{\mbox{\bf v}}$. Thus $HF^+(-Z_0(L), {\mathfrak{s}}_{1-g})$ is generated by the homology class of ${\mbox{\bf u}}$.

-   There is a unique holomorphic triangle $\psi$ contributing to the image of ${\mbox{\bf u}}$ under the chain map $F_W$.

-   The image of ${\mbox{\bf u}}$ is a cycle representing the contact invariant $c(\xi)\in HF^+(-Z)$. More precisely, the image 