\CFa(Y,\spinc),$$ with the summand $\CFa(Y,\spinc)$ generated by those $\x\in\mathcal{G}$ with $\spinc_w(\x)=\spinc$. Using both basepoints,  define a map: $$\underline{\spinc}:=\spinc_{w,z}: \mathcal{G}\rightarrow \underline{\SpinC}(Y,K),$$ where $ \underline{\SpinC}(Y,K)$ is the set of relative $\SpinC$ structures on $Y-K$. Picking a Seifert surface, $F$, for $K$, we obtain a map: $$A : \mathcal{G}\rightarrow \Z,$$ by associating the quantity $A(\x)=\OneHalf\langle c_1(\spincrel(\x)),[F,\partial F]\rangle $ to a generator. Here $c_1(\spincrel(\x))\in H^2(Y,K;\Z)$ is the relative first Chern class of the relative $\SpinC$ structure. We will refer to $A$ as the [*Alexander grading*]{}. $A$ defines a filtration on $\CFa(Y,\spinc)$, in the sense that $A(\partial (\x))\le A(\x)$ for each $\x\in \mathcal{G}$. Note that $A$ depends on the Seifert surface, but only through its homology class.

Thus, once we pick a homology class of Seifert surface, it makes sense to define $\FiltY(K,i)$ to be the subcomplex of $\CFa(Y,\spinc)$ generated by $\x\in \mathcal{G}$ satisfying $A(\x)\le i$. We then have the finite length filtration mentioned in the introduction: $$0=\FiltY(K,-i) \subseteq \FiltY(K,-i+1)\subseteq \ldots \subseteq
\FiltY(K,n)=\CFa(Y,\spinc).$$ We will habitually omit the Seifert surface from the notation and assume throughout that we have chosen a fixed Seifert surface, drawing attention to its role only when there may be some ambiguity.

Finally, we recall the following definition from [@tbbounds]. To 