of the contact submodule implies tightness (c.f. [@Contact Theorem 1.4]).

Suppose a contact structure $\xi$ is overtwisted. Then the contact submodule is trivial, i.e. ${\underline}{c}(\xi;\mathbb{A})\equiv 0$ for any ground module $\mathbb{A}$.

This follows in exactly the same manner as [@Contact Proof of Theorem 1.4], noting only that the K[ü]{}nneth theorem for the knot Floer filtration of the connected sums of knots holds with arbitrary coefficient modules and that the contact submodule associated to the overtwisted contact structure induced by the left-handed trefoil is trivial over any ground module (this latter fact can be calculated directly or deduced from the universal coefficients theorem and the fact that ${{\mathbb F}}[H^1(S^3)]={{\mathbb F}}$).

The preceding two lemmas yield an immediate corollary.

In the situation of Lemma \[fulltwistlemma\], and for $g\geq 2$, if the map $$F^{{{\mathbb F}}[C_n]}_W: {\underline}{HF}^+(-Z_0(K), {\mathfrak{s}}_{1-g}; {{\mathbb F}}[C_n])\to {\underline}{HF}^+(-Z, {\mathfrak{s}}_\xi; {{\mathbb F}}[C_n])$$ is nonzero, then $\xi_L$ is tight. If $g=1$ and the map $$F^{\Lambda_\omega[C_n]}_{W}: {\underline}{HF}^+(-Z_0(K), {\mathfrak{s}}_{1-g}; \Lambda_\omega[C_n])\to {\underline}{HF}^+(-Z, {\mathfrak{s}}_\xi; \Lambda_\omega[C_n])$$ is nonzero, then $\xi_L$ is tight.

We now return to the surgery exact triangle. In this case, the manifold $Z$ above becomes $Y_{-1/n}(K)$, while $Z_0 = Y_0(K)$. We have:

\[tightprop\] Let $n>0$, and suppose the genus of the fiber is greater than 