that $K_{p,pn+1}$ is a $\C$-knot. Then $n\ge -2(\frac{\tau(K)}{p-1}+ \frac{1}{p})$.

\[cor:cknot\] Suppose $K_{p,pn+1}$ is a $\C$-knot with defining complex curve, $V_f$. Further, suppose the genus of the piece of $V_f$ contained in the four-ball is equal to the Seifert genus, $g(K)$. Then $n\ge 0$ and $\tau(K)=g(K)$.

Thus, for instance, no negative cable (i.e. $n<0$) of a knot with $\tau(K)<0$ (e.g. the left-handed trefoil) will ever be a $\C$-knot. Note, too, that since $\tau(K)\le g_4(K)$, Corollary \[cor:cor1\] could also be phrased as a (weaker) obstruction which depends solely on the smooth four-genus of $K$.

Another feature of links of singularities is that they are fibered and, as alluded to above, there is an isotopy taking their fiber surface to their Milnor fiber. Restricting to the category of fibered knots whose fiber surface is isotopic to a piece of a complex curve, we have the following characterization theorem:

Corollary $1.4$ of [@ComplexCable]\[thm:ComplexCable\] Let $K$ be a fibered knot. Then $K_{p,q}$ has a Seifert surface which is isotopic to a piece of a complex curve $V_f \cap B^4$ if and only if

-   $K$ has a Seifert surface which is isotopic to a piece of a complex curve [*and*]{}

-   $q>0$

In particular, the 