of whether we use the basepoint $z$ or $z^\prime$. The only restriction on the differential for the filtered subcomplexes is that $n_w(\phi)=0$, which is independent of $z$ and $z^\prime$. In light of these remarks and the table of filtrations, we have the following isomorphisms of [*chain complexes*]{}, for each $\spinc\in \SpinC(Y)$ and $k<l$ $$\begin{array}{ll} \FiltY(K_{p,pn+1},pg+\frac{(pn)(p-1)}{2}-p(k-1)-1))& = \FiltY(K,g-k)   \\
    \FiltY(K_{p,pn+1},pg+\frac{(pn)(p-1)}{2}-p(k-1)-i)& = \FiltY(K_{p,pn+1},pg+\frac{(pn)(p-1)}{2}-p(k-1)-i-1)) \\
&   \forall \ i=2,\ldots,p-1  \\ \end{array}$$

Note the appearance of the $\SpinC$ structure, $\spinc$. Up to this point we had not distinguished between intersection points corresponding to different $\SpinC$ structures; indeed, the table represents all intersection points. However, it is straightforward to see that the table splits as a direct sum of complexes according to $\SpinC$ structures on $Y$, yielding the above. This follows from the fact that part $(1)$ of Lemma \[lemma:pindown\] actually applies to any exterior point in the table which, in turn, follows from the fact that $\{x_j,\y\}$ is connected to $\{x_0,\y\}$ by a Whitney disk (c.f the proof of Lemmas $3.3$ and $3.4$ of [@Cabling]).

Taking $l>g-M$ in Lemma \[lemma:isolate\] and changing variables $j-1=g-k$ yields the first part of the theorem. For the second part we let $M<-g(K)$, where $g(K)$ is the genus of 