$n$-framed longitude for the knot $(p-1)$ times. $w$ is to remain fixed under this operation. $z$ is replaced by a basepoint $z'$ so that the arc connecting $z'$ and $w$ has intersection number $p$ with $\tilde{\beta}$. See Figure  \[fig:wind\].

![\[fig:wind\] Shown are parts of the diagrams $H$ and $H(p,n)$ (here, $p=3$ and $n=2$). On $H(p,n)$, we have slid the meridian, $\mu$, $2$ times along a $2$-framed longitude $\lambda$. This figure establishes the ordering on exterior intersection points in terms of the ordering on $x_i\in \alpha_g\cap \tilde{\beta}$. ](winding.eps)

Lemma 3.1 of [@Cabling] shows that with the addition of a third basepoint (which, by an abuse of notation, we also denote by $z$) $H(p,n)$ specifies $K$ by using the pair $(w,z)$ while it specifies $K_{p,pn+1}$ using the pair $(w,z^\prime)$. See Figure \[fig:wind\]. Increasing the parameter $n$ for the cable has the effect of adding many generators to the knot Floer chain complexes derived from $H(p,n)$. These generators, $\x'$, were called [*exterior intersection points*]{} in the predecessor, and are characterized by the property of having one component of $\x'$ of the form $x_i\in \alpha_g\cap \tilde{\beta}$, with $x_i$ lying in a small neighborhood of the meridian, $\mu$. The exterior points are in $(2n(p-1)+1)$-to-one correspondence 