knot in $Z$; namely, the core of the surgery solid torus used to obtain $Z$ as $-1/n$ surgery on $K\subset Y$. Moreover, this knot is fibered in $Z$ with monodromy differing from that of $K$ by $n$ right-handed boundary Dehn twists, and thus it induces the contact structure we had been calling $\xi_n$ on $Z=Y_{-1/n}(K)$. While this is the application we have in mind, for the moment we suppress the auxiliary 3-manifold, fibered knot, and contact structure, $(Y,\xi,K)$, and simply consider the general case of a fibered knot $L\subset Z$ inducing a contact structure which we abusively denote by $\xi$, and the associated zero surgery $Z_0$.

We will need a generalization of the Heegaard Floer contact invariant introduced in [@Contact] to the situation of twisted coefficients. This generalization is alluded to in [@Contact Remark 4.5] and further developed in [@GenusBounds Section 4]. The construction associates to a contact structure $\xi$ on $Z$ and any module $\mathbb{A}$ over ${{\mathbb F}}[H^1(Z)]$, a distinguished submodule: $${\underline}{c}(\xi;\mathbb{A}):=\iota(\mathbb{A})\subset {\underline}{HF}^+(-Z;\mathbb{A}).$$ This [*contact submodule*]{} is generated by the inclusion $\iota$ of the homology of the “bottommost" non-trivial filtered submodule of the knot Floer homology of a fibered knot supporting $\xi$ (which is isomorphic to $\mathbb{A}$ by [@Contact 