of ${\mbox{\bf u}}$ is the unique generator for the knot Floer chain complex for $L$, in filtration level $-g$.

Using the same diagram for the chain complexes and chain maps with twisted coefficients gives the desired result. Indeed, all of the statements above remain true with $\mathbb{A}$ replacing the implicit ${{\mathbb F}}$ coefficients. Note that the second item in this case establishes an isomorphism $HF^+(-Z_0(L), {\mathfrak{s}}_{1-g};\mathbb{A})\cong \mathbb{A}$ as a trivial ${{\mathbb F}}[U]$-module, but generation by ${\mbox{\bf u}}$ is ambiguous; in particular, it does [*not*]{} mean generation as an ${{\mathbb F}}[H^1(Z_0)]$-module since $\mathbb{A}$ may not even be finitely generated over ${{\mathbb F}}[H^1(Z_0)]$. This is, in essence, why one needs to talk about the contact [*submodule*]{} rather than the contact [*element*]{} in the most general case.

When $g = 1$, the key difference is that $c_1({\mathfrak{s}}_{1-g})$ is torsion and the diagram fails to be admissible. However, it fails to be admissible only because of positivity of the periodic domain corresponding to the homology class of the fiber. If we take coefficients in an algebra over $\Lambda_\omega$, where $\omega$ evaluates non-trivially on this class, then no admissibility is required for this periodic domain.

Just as with contact element in untwisted Floer homology, non-vanishing 