with the generators of the chain complex for $K$ which came from $H$, i.e. there is $(2n(p-1)+1)$-to-one map: $$\pi:\{\mathrm{exterior \ intersection \ points \ of \ H(p,n)} \}\longrightarrow \{\mathrm{intersection \ points \ of\ H}\}.$$

More precisely, any generator for the Heegaard diagram $H$ is of the from $\{p,\y\}$, where $p\in \alpha_g\cap \mu$ and $\y$ is a $(g-1)$-tuple of intersection points. The fibers of $\pi$ are given by $$\pi^{-1}(\{p,\y\}) = \{ \ \{x_i,\y\} \ |  \ x_i\in \alpha_g\cap \tilde{\beta}, \ i=0,..., 2n(p-1)\}.$$ Figure \[fig:wind\] establishes an ordering for the fiber $\pi^{-1}(\{p,\y\})$, in terms of an ordering on the $x_i$. In terms of this ordering, we call intersection points of the form $\{x_0,\y\}$ [*outermost intersection points*]{}. Let us establish some notation: $$C(i):= \{ (g-1)\sd \text{tuples},\ \y \ | \ \{p,\y\} \text{\ has\ Alexander\ grading\ } i \text{\ for\  }H  \}$$ $$A:= \text{\ Alexander\ grading\ for\ generators\ from\   } H(p,n),\text{\ with\ respect\ to\ } (w,z)$$ $$A':= \text{\ Alexander\ grading\ for\ generators\ from\        } H(p,n),\text{\ with\ respect\ to\  } (w,z')$$

We will determine the Alexander gradings, $A$ and $A'$, of exterior intersection through a sequence of lemmas. The first is an adaptation of Lemma $3.6$ of [@OSThurston] to the present notation. It determines $A$ 