Proof of Theorem 1.1]) into the Floer homology of $-Z$. Strictly speaking, the literature only refers to the contact [*element*]{} in twisted Floer homology, but this does not make sense with coefficients in a general module.

The contact submodule behaves well with respect to the $2$-handle cobordism described above, a fact which we now make precise. To state the result, note that there is a canonical [[ ]{}]{}structure ${\mathfrak{s}}_{1-g}$ on $Z_0$ determined by

-   ${\mathfrak{s}}_{1-g}$ is cobordant through the surgery cobordism to the [[ ]{}]{}structure on $Z$ determined by the contact structure,

-   If ${\widehat{S}}$ denotes the fiber of the open book, capped off in $Z_0$, then we have: $$\label{eq:extremal}
    \langle c_1({\mathfrak{s}}_{1-g}), [{\widehat{S}}]\rangle = 2-2g.$$

\[fulltwistlemma\] Let $L\subset Z$ be a fibered knot with induced contact structure $\xi_L$. If the fiber of $L$ has genus greater than one, then for any module $\mathbb{A}$ over ${{\mathbb F}}[H^1(Z_0(L))]$ there is an identification $${\underline}{HF}^+(-Z_0(L),{\mathfrak{s}}_{1-g};\mathbb{A}) \cong  \mathbb{A}$$ as a trivial ${{\mathbb F}}[U]$-module, i.e. $U$ acts as zero. Moreover, ${\mathfrak{s}}_{1-g}$ is the unique ${{\mbox{spin$^c$} }}$-structure satisfying and supporting non-zero Floer homology. The image of the map $${\underline}{F}_{W}^{\mathbb{A}}: {\underline}{HF}^+(-Z_0(L), {\mathfrak{s}}_{1-g};\mathbb{A})\to {\underline}{HF}^+(-Z, {\mathfrak{s}}_\xi;\mathbb{A})$$ induced by the 0-surgery cobordism is the contact submodule ${\underline}{c}(\xi_L;\mathbb{A})$.

All of the above 