\section{Cables}\label{sec:cables}
Let $K$ be a knot in $S^3$, with regular neighborhood $N(K)$.  Let $C$ be a simple loop on $\partial N(C)$ with slope $p/q$, where $q\ge 2$.  Then $C$ is the {\em $(p,q)$-cable} of $K$, $C_{p,q}(K)$.  If $K$ is trivial then $C_{p,q} (K)$ is the torus knot $T_{p,q}$.  Since $C_{-p,q} (K) = C_{p,-q}(K) = - C_{p,q}(-K)$, we will always assume that $p \geq1$ and $q\ge 2$.

We recall the main theorem about cables that we are interested in proving.  \\ \\
{\bf Theorem~\ref{thm:cables}.}  {\em Let $K$ be a non-trivial knot in $S^3$.  Then, $\Sigma_n(C_{p,q}(K))$ is excellent, unless $n = q = 2$.} \\

In the case $n=q=2$, it turns out that $\Sigma_2 (C_{p,2} (K)) \cong X_1 \cup_\partial X_2$, where $X_i$ is a copy of the exterior $X$ of $K$, $i=1,2$.  For $p=1$ we get the following conditional result.

\begin{theorem}\label{thm:conditional}
If $\frac{1}{2}$ is a CTF slope for $K$ then $\Sigma_2 (C_{1,2} (K))$ is excellent.
\end{theorem}
In particular, if the Li-Roberts Conjecture holds, then $\Sigma_2(C_{1,2}(K))$ is excellent for all non-trivial $K$.  The theorem will be proved at the end of the section.  

Let $S^3 = V\cup_T W$ be the standard genus~1 Heegaard splitting of $S^3$.  Let $C\subset \text{int }W$ be an isotopic copy of a $(p,q)$-curve on $T$, where this curve intersects the meridional disk of $W$ $q$ times.  Let $J$ be a core of $V$.  Let $\sigma,\tau \subset T$ be meridians of $V,W$, respectively.  We will be more precise about our orientations of these curves shortly.

In $S^3 = V\cup W$ the curve $C$ is the torus knot $T_{p,q}$.  In the corresponding Seifert fibration of $S^3$, in which $C$ is an ordinary fiber, $J$ is 
the exceptional fiber of multiplicity $p$.  Let $\pi : \Sigma_n (T_{p,q}) \to S^3$ be the branched covering projection.  Then $\pi^{-1}(V)$ has $\gcd(n,q)$ components $\widetilde V_i$, $1\le i\le \gcd(n,q)$, each an $\frac{n}{\gcd(n,q)}$-fold covering of $V$.  In the induced Seifert fibration on $\Sigma_n (T_{p,q})$ the core of each $\widetilde V_i$ is an exceptional fiber of multiplicity $\frac{p}{\gcd(n,p)}$. Let $\widetilde \sigma_i$ be a meridian of $\widetilde V_i$, $1\le i\le \gcd(n,q)$. Also, $\widetilde\tau_i = \pi^{-1} (\tau) \cap \partial\widetilde V_i$ is connected for $1\le i\le \gcd(n,q)$.  Let $s, \varphi \subset \partial W$ be an oriented section-fiber pair for the induced Seifert structure on $W$ (i.e. $s \cdot \varphi = 1$).  We orient $\tau$ such that $\tau \cdot \varphi = q$ in $\partial W$.  Observe that $\varphi$ is a $(p,q)$-curve on $\partial W$.  We then orient $\sigma$ such that $\tau \cdot \sigma = 1$ on $\partial W$.    


Let $X$ be the exterior of the knot $K$, and let $\lambda,\mu$ be a ($0$-framed) longitude-meridian pair on $\partial X$.  If we remove $int(V)$ from $S^3$ and replace it by $X$, identifying $\partial X$ with $\partial W$ in such a way that $\lambda,\mu$ are identified with $\sigma,\tau$, respectively, we get $S^3$, and the curve $C$ becomes the $(p,q)$-cable of $K$.  


Let $\widetilde W = \pi^{-1}(W) = \Sigma_n (T_{p,q}) - \coprod_{i=1}^{\gcd(n,q)} int(\widetilde V_i)$, the $n$-fold cyclic branched cover of $W$ branched along $C$. 
Let $\widetilde X$ be the $\frac{n}{\gcd(n,q)}$-fold cyclic covering of $X$, and let $\widetilde\lambda, \widetilde\mu \subset \partial \widetilde X$ be as in the discussion in Section~\ref{subsec:branchedcovers}.  Then $\Sigma_n (C_{p,q}(K)) = \widetilde W \cup \coprod_{i=1}^{\gcd(n,q)} \widetilde X_i$, where $\widetilde X_i$ is a copy of $\widetilde X$ and $\partial\widetilde X_i$ is glued to $\partial\widetilde V_i$ in such a way that $\widetilde\lambda_i,\widetilde \mu_i$ are identified with $\widetilde\sigma_i,\widetilde\tau_i$, respectively.

\subsection{The proof of Theorem~\ref{thm:cables}}
The following proposition establishes Theorem~\ref{thm:cables} in the generic case.


\begin{proposition}\label{prop:generic}
Let $K$ be a knot in $S^3$.  Suppose $p\ne 1$ and $\{ n,p,q\} \ne \{2,2,r\}$, $\{ 2,3,3\}$, $\{2,3,4\}$ or $\{2,3,5\}$.  Then $\Sigma_n (C_{p,q}(K))$ is excellent.
\end{proposition}
\begin{proof}
Using the notation in the discussion before the statement of the proposition, $\widetilde W(\widetilde{\boldsymbol\sigma}) = \widetilde W (\widetilde\sigma_1,\ldots, \widetilde\sigma_{\gcd(n,q)}) = \Sigma_n (T_{p,q})$, which by Theorem~\ref{thm:torus} is excellent if  $\{n,p,q\}$ is not one of the exceptions listed in the proposition. Hence by Lemma~\ref{lem:seifertboundary}, $\widetilde{\boldsymbol\sigma}$ is excellent for $\widetilde W$. By \cite{Gabai3}, $\lambda$ is a CTF slope for $X$, and hence $\widetilde\lambda$ is a CTF  slope for $\widetilde X$.  In particular, we have that $\widetilde X(\widetilde\lambda)$ is prime. Since $H_1(X(\lambda))$ is infinite, so is $H_1(\widetilde X(\widetilde\lambda))$.  Therefore, $\widetilde \lambda$ is an LO slope for  $\widetilde X$ by \cite[Corollary 3.4]{BRW}.  Thus $\Sigma_n (C_{p,q}(K))$ is excellent by Lemma~\ref{lem:excellentgluing}.
\end{proof}

To treat the cases where $(n,p,q)$ is one of the exceptional triples in Theorem~\ref{thm:torus} we continue with a more detailed analysis of the description of $\Sigma_n(C_{p,q}(K))$ given just before the present subsection.  

Recall that $\varphi \subset \partial W$ is a regular fiber in the Seifert fibration of $S^3$ described above.  Note that $\widetilde\varphi_i = \pi^{-1}(\varphi) \cap \partial \widetilde{V}_i$ has $\gcd(\frac{n}{\gcd(n,q)},p) = \gcd(n,p)$ components.  Since we have oriented $\varphi$ so that $\tau \cdot \varphi = q$ on $\partial W$, we have $\widetilde{\tau}_i \cdot \widetilde{\varphi_i} = \frac{nq}{\gcd(n,q)\gcd(n,p)}$ on $\partial \widetilde{W}$.  For notation, we let $\omega = \frac{nq}{\gcd(n,q)\gcd(n,p)}$.    

We have $\widetilde{W}(\widetilde{\sigma}_1,\ldots,\widetilde{\sigma}_{\gcd(n,q)}) = \Sigma_n(T_{p,q})$.  As in Section~\ref{sec:torus}, we will use the description of the Seifert invariants of cyclic branched covers of torus knots in \cite[Theorem 1]{NR}.  We define $\widetilde{\gamma}_i \subset \partial \widetilde{V}_i$ as the section with respect to which the Seifert invariants of $\Sigma_n(T_{p,q})$ are described in \cite[Theorem 1]{NR}.  Thus $\widetilde{\gamma}_i \cdot \widetilde{\varphi}_i = 1$ in $\partial \widetilde{W}$, and if the core of $\widetilde{V}_i$ has Seifert invariants $\frac{r}{s}$ in $\Sigma_n(T_{p,q})$, then $\widetilde{\sigma}_i = \pm (s\widetilde{\gamma}_i + r\widetilde{\varphi}_i)$.  In some cases, the orientation of $\widetilde{\sigma}_i$ will be  easily determined, while often it will not.  In the latter situation, rather than repeat the constructions of \cite{NR} in precise detail to obtain the exact orientation of $\widetilde{\sigma}_i$, it will be easier to simply treat both cases, even though only one can possibly arise.      

Observe that $\widetilde{\tau}_i = \omega \widetilde{\gamma}_i + \eta \widetilde{\varphi}_i$ for some $\eta \in \mathbb{Z}$.  Recall that we must have that $\widetilde{\tau}_i \cdot \widetilde{\sigma}_i = \widetilde{\gamma}_i \cdot \widetilde{\varphi}_i = 1$ on $\partial \widetilde{W}$.  This will determine the possible values of $\eta$.  

The strategy to complete the proof of Theorem~\ref{thm:cables} is as follows.  We consider slopes $\alpha_k = \mu + k\lambda$ on $\partial X$, $k \in \Z$.  The corresponding slope on $\partial \widetilde{X}_i$ is $\widetilde{\alpha}_k = \widetilde{\mu}_i + \frac{n}{\gcd(n,q)} k \widetilde{\lambda}_i$, which is identified with $\ttau_i + \frac{n}{\gcd(n,q)} k \widetilde{\sigma}_i$ on $\partial \widetilde{W}$.  We will show that for $k \gg 0$ or $k \ll 0$,  $\widetilde{W}(\widetilde{\alpha}_k,\ldots,\widetilde{\alpha}_k)$ is a Seifert fibered space with a horizontal foliation, and therefore $(\widetilde{\alpha}_k,\ldots,\widetilde{\alpha}_k)$ is an excellent multislope for $\widetilde{W}$ by Lemma~\ref{lem:seifertboundary}.  Since, for $|k|$ sufficiently large, $\widetilde{\alpha}_k$ is an excellent slope for $\widetilde{X}$ by Lemma~\ref{lem:cover1/n}, $\Sigma_n(C_{p,q}(K))$ will be excellent by Lemma~\ref{lem:excellentgluing}.  

The details are given in Propositions~\ref{prop:exceptional}, \ref{prop:234}, and \ref{prop:1q} below.  

We first treat the case where $\gcd(n,q) = 1$ and $p \neq 1$.  Then, $\pi^{-1}(V) = \widetilde{V}$ is connected, and we drop the subscript $i$ from $\widetilde{V}_i, \widetilde{\sigma}_i$, etc.  Also, for the rest of this section, $X^{(n)}$ will denote the $n$-fold cyclic covering of $X$.  

\begin{proposition}\label{prop:exceptional}
Let $K$ be a non-trivial knot in $S^3$.  If $\{n,p,q\} = \{2,3,5\}$ or $(n,p,q) = (2,4,3), (3,3,2)$, $(4,2,3)$, or $(2,2,q)$, then $\Sigma_n(C_{p,q}(K))$ is excellent.  
\end{proposition}

\begin{proof}
Note that $\widetilde{J} = \pi^{-1}(J)$ is a fiber of multiplicity $\frac{p}{\gcd(n,p)}$ in $\Sigma_n(T_{p,q})$.  Also, $\widetilde{\tau} = \omega  \widetilde{\gamma} + \eta \widetilde{\varphi}$ where $\omega = \frac{nq}{\gcd(n,p)}$, since $\gcd(n,q) = 1$.  Here, $\talpha_k = \tmu + nk\tlambda$ is identified with $\ttau + nk\tsigma$ on $\partial \widetilde{W}$.  Thus, by the discussion preceding the statement of the proposition, it suffices to show that $\widetilde{W}(\ttau + nk \tsigma)$ admits a horizontal foliation for $k \gg 0$ or $k \ll 0$.  \\

\noindent {\bf Case 1:} $(n,p,q) = (2,3,5)$.\\
From \cite[Theorem 1]{NR}, $\Sigma_2(T_{3,5}) = M(1,-\frac{1}{2},-\frac{1}{3},-\frac{1}{5})$.  Since $\widetilde{J}$ is the fiber of multiplicity 3, $\widetilde{\sigma} = \pm(-3\widetilde{\gamma} + \widetilde{\varphi})$.  We have $\omega = 10$, and thus 
\[
1 = \ttau \cdot \tsigma = \pm (10 \tgamma + \eta \tphi) \cdot (-3 \tgamma + \tphi) = \pm (3\eta + 10).
\]
Therefore, we have that $\eta = -3$, and thus $\widetilde{\tau} = 10 \widetilde{\gamma} - 3\widetilde{\varphi}$ and $\widetilde{\sigma} = -3\tgamma + \tphi$.  

The slope $\widetilde{\alpha}_k = \widetilde{\mu} + 2 k \widetilde{\lambda}$ on $\partial X^{(2)}$ is identified with $\widetilde{\tau} + 2  k \widetilde{\sigma} = (10 \widetilde{\gamma} - 3 \widetilde{\varphi}) + 2k( -3 \widetilde{\gamma} + \widetilde{\varphi}) = (10-6k) \widetilde{\gamma} + (-3+2k)\widetilde{\varphi}$ on $\partial \widetilde{W}$.  Hence $\widetilde{W}(\widetilde{\alpha}_k)$ has Seifert invariants $M(1,-\frac{1}{2},-\frac{1}{5}, \frac{2k-3}{10-6k}) = -M(-1,\frac{1}{2},\frac{1}{5}, \frac{2k-3}{6k-10})$.  If $k \leq 0$, then $0 < \frac{2k-3}{6k-10} < \frac{1}{3}$.  By choosing $m = 3$ and $a = 1$, it follows from Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition} that $\widetilde{W}(\widetilde{\alpha}_k)$ has a horizontal foliation.  This is what we wanted to show.  \\

\noindent {\bf Case 2:} $(n,p,q) = (2,5,3)$. \\
 Here $\widetilde{\sigma} = \pm (-5 \widetilde{\gamma} + \widetilde{\varphi})$, and $\widetilde{\tau} = 6 \widetilde{\gamma} + \eta \widetilde{\varphi}$.  We have 
\[
1 = \ttau \cdot \tsigma = \pm (6 \tgamma + \eta \tphi) \cdot (-5\tgamma + \tphi) = \pm (6 + 5 \eta).  
\]
Therefore, $\eta = -1$ and we have $\ttau = 6 \tgamma - \tphi$, $\tsigma = -5 \tgamma + \tphi$.  
Hence on $\partial \widetilde{W}$, $\widetilde{\alpha}_k = \widetilde{\tau} + 2k \tsigma = (6-10k)\tgamma + (2k-1)\tphi$.  Then $\widetilde{W}(\talpha_k)$ is the Seifert fibered space $M(1,-\frac{1}{2},-\frac{1}{3}, \frac{2k-1}{6-10k}) = - M(-1,\frac{1}{2},\frac{1}{3},\frac{2k-1}{10k-6})$.  For $k \leq 0$, $0 < \frac{2k-1}{10k-6} < \frac{1}{5}$.  Letting $m = 5$ and $a = 2$, we see that $\widetilde{W}(\talpha_k)$ has a horizontal foliation by Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}. \\

\noindent {\bf Case 3:} $(n,p,q) = (3,2,5)$.\\
 Here $\tsigma = \pm (-2 \tgamma + \tphi)$, and $\ttau = 15 \tgamma + \eta \tphi$.  In this case, we have 
\[
1 = \pm (15 \tgamma + \eta \tphi) \cdot (-2 \tgamma + \tphi) = \pm (15 + 2 \eta).  
\]
Thus, we have two possibilities.  The first is that $\eta = -7$, $\ttau = 15 \tgamma - 7 \varphi$, and $\tsigma = -2\tgamma + \tphi$.  The second case is that $ \eta = -8$, $\ttau = 15\tgamma - 8 \tphi$, and $\tsigma = 2 \tgamma - \tphi$.  

First, suppose that $\eta = -7$.  The slope $\talpha_k = \tmu + 3k \tlambda$ on $\partial X^{(3)}$ is identified with $\ttau + 3k \tsigma = (15-6k)\tgamma + (3k-7)\tphi$ on $\partial \tW$.  Therefore $\tW(\talpha_k)$ has Seifert invariants  $M(1,-\frac{1}{3},-\frac{1}{5}, \frac{3k-7}{15-6k}) = -M(-1,\frac{1}{3},\frac{1}{5},\frac{3k-7}{6k-15})$.  If $k \leq 0$, then $0< \frac{7-3k}{15-6k} < \frac{1}{2}$ and $\tW(\talpha_k)$ has a horizontal foliation by Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}. 

Next, suppose that $\eta = -8$.  The slope $\talpha_k = \tmu + 3k \tlambda$ on $\partial X^{(3)}$ is identified with $\ttau + 3k \tsigma = (15+6k)\tgamma + (-8-3k)\tphi$ on $\partial \tW$.  Therefore $\tW(\talpha_k)$ has Seifert invariants  $M(1,-\frac{1}{3},-\frac{1}{5}, \frac{-3k-8}{6k+15}) = -M(-1,\frac{1}{3},\frac{1}{5},\frac{3k+8}{6k+15})$.  If $k \ll 0$, then $0 < \frac{3k+8}{6k+15} < \frac{1}{2}$ and $\tW(\talpha_k)$ has a horizontal foliation by Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}. 

  

The cases $(n,p,q) = (3,5,2), (5,2,3)$, and $(5,3,2)$ are completely analogous; we leave the details to the reader.  \\

\noindent {\bf Case 4:} $(n,p,q) = (2,4,3)$.  \\
By \cite[Theorem 1]{NR}, $\Sigma_2(T_{4,3}) = M(\frac{1}{2},-\frac{1}{3},-\frac{1}{3})$.  Since $\widetilde{J}$ has multiplicity 2, we have $ \tsigma = \pm(2 \tgamma  + \tphi)$.  Also, $\omega = \frac{nq}{\gcd(n,p)} = 3$.  We again have two cases.  The first case is $\eta = 1$, $\ttau = 3 \tgamma + \tphi$, $\tsigma = 2 \tgamma + \tphi$.  The slope $\talpha_k = \tmu + 2k\tlambda$ on $\partial X^{(2)}$ is identified with $\ttau + 2k \tsigma = (4k+3)\tgamma + (2k+1)\tphi$ on $\partial \tW$.  So 
\[
\tW(\talpha_k) = M\left(-\frac{1}{3},-\frac{1}{3}, \frac{2k+1}{4k+3}\right) = -M\left(\frac{1}{3},\frac{1}{3},-\frac{2k+1}{4k+3}\right) = -M\left(-1,\frac{1}{3},\frac{1}{3},\frac{2k+2}{4k+3}\right).
\]
Since $\frac{2k+1}{4k+3} < \frac{1}{2}$ for $k \geq 0$, $\tW(\talpha_k)$ has a horizontal foliation by Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}.  

The second case is $\eta = 2$, $\ttau = 3 \tgamma + 2\tphi$, $\tsigma = -2 \tgamma -\tphi$.  The slope $\talpha_k = \tmu + 2k\tlambda$ on $\partial X^{(2)}$ is identified with $\ttau + 2k \tsigma = (3 - 4k)\tgamma + (2-2k)\tphi$ on $\partial \tW$.  So 
\[
\tW(\talpha_k) = M\left(-\frac{1}{3},-\frac{1}{3}, \frac{2k-2}{4k-3}\right) = -M\left(\frac{1}{3},\frac{1}{3},\frac{2-2k}{4k-3}\right) = -M\left(-1,\frac{1}{3},\frac{1}{3},\frac{2k-1}{4k-3}\right).  
\]
Since $\frac{2k-1}{4k-3} < \frac{1}{2}$ for $k \leq 0$, $\tW(\talpha_k)$ has a horizontal foliation by Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}.  \\


\noindent {\bf Case 5:} $(n,p,q) = (3,3,2)$.  \\
By \cite[Theorem 1]{NR}, $\Sigma_3(T_{3,2}) = M(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{1})$.  Since $\widetilde{J}$ has multiplicity 1, we have $\tsigma = \pm (\tgamma + \tphi)$.  We have two cases.  

First, $\ttau = 2 \tgamma + \tphi$, $\tsigma = \tgamma + \tphi$.  The slope $\talpha_k = \tmu + 3k \tlambda$ on $\partial X^{(3)}$ is identified with $\ttau + 3k \tsigma = (3k+2) \tgamma + (3k+1) \tphi$ on $\partial \tW$.  Therefore, $\tW(\talpha_k)$ has Seifert invariants 
\[
M\left(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{3k+1}{3k+2}\right) = M\left(-2,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{-1}{3k+2}\right). 
\]
For $k \ll 0$, the latter are normalized Seifert invariants, and thus $\tW(\alpha_k)$ has a horizontal foliation by Condition~\eqref{condition:fibers} of Theorem~\ref{thm:brwcondition}.   

Second, we have $\ttau = 2\tgamma + 3\tphi$, $\tsigma = -\tgamma - \tphi$.  The slope $\talpha_k = \tmu + 3k \tlambda$ on $\partial X^{(3)}$ is identified with $\ttau + 3k \tsigma = (2-3k) \tgamma + (3-3k) \tphi$ on $\partial \tW$.  Therefore, $\tW(\talpha_k)$ has Seifert invariants 
\[
M\left(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{3-3k}{2-3k}\right) = M\left(-2,\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2-3k}\right).
\]
For $k \leq 0$, the latter invariants are normalized, and thus $\tW(\alpha_k)$ has a horizontal foliation by Condition~\eqref{condition:fibers} of Theorem~\ref{thm:brwcondition}.   \\


\noindent {\bf Case 6:} $(n,p,q) = (4,2,3)$.\\
 By \cite[Theorem 1]{NR} $\Sigma_4(T_{2,3}) = M(-\frac{1}{2},\frac{1}{1},-\frac{1}{3},-\frac{1}{3})$.  Since $\widetilde{J}$ has multiplicity 1, we have $ \tsigma = \pm (\tgamma + \tphi)$.  Again, there are two cases.  
 
For the first case, we have $\ttau = 6 \tgamma + 5 \tphi$, $\tsigma = \tgamma + \tphi$.  The slope $\talpha_k = \tmu + 4k \tlambda$ on $\partial X^{(4)}$ is identified with $\ttau + 4k \tsigma = (4k+6) \tgamma + (4k+5) \tphi$ on $\partial \tW$.  Thus, $\tW(\alpha_k)$ has Seifert invariants
 \[
 M\left(-\frac{1}{2},-\frac{1}{3},-\frac{1}{3},\frac{5+4k}{6+4k}\right) = -M\left(-2,\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{4k+7}{4k+6}\right).  
\]
Hence for $k \leq -2$ we are in Condition~\eqref{condition:fibers} of Theorem~\ref{thm:brwcondition} and $\tW(\talpha_k)$ has a horizontal foliation. 

For the second case, we have $\ttau = 6 \tgamma + 7 \tphi$, $\tsigma = -\tgamma - \tphi$.  In this case, $\tW(\talpha_k)$ has Seifert invariants
\[
 M\left(-\frac{1}{2},-\frac{1}{3},-\frac{1}{3},\frac{7-4k}{6-4k}\right) = -M\left(-2,\frac{1}{2},\frac{1}{3},\frac{1}{3},\frac{4k-5}{4k-6}\right).  
\]
Hence for $k \leq 0$ we are in Condition~\eqref{condition:fibers} of Theorem~\ref{thm:brwcondition} and $\tW(\talpha_k)$ has a horizontal foliation. \\


\noindent {\bf Case 7:} $(n,p,q) = (2,2,q)$. \\
Observe that $\Sigma_2(T_{2,q})$ is a lens space.  More specifically, from \cite[Theorem 1]{NR} we get that $\Sigma_2 (T_{2,q})$ has Seifert invariants $M(-\frac11, \frac{\beta_2}q, \frac{\beta_2}q)$ where $\beta_2 =\frac{q-1}2$.  Since $J$ has multiplicity~2, and $\widetilde J\to J$ is a 2-fold connected covering, we have $\widetilde J$ is the fiber with Seifert invariant $-\frac{1}1$.  
Hence $\widetilde\sigma = \pm(-\widetilde\gamma + \widetilde\varphi)$.  Again, there are two cases.  

First, we have $\ttau = q \tgamma + (1-q) \tphi$ and $\tsigma = -\tgamma + \tphi$.  In $\Sigma_2 (C_{2,q}(K)) \cong X^{(2)} \cup \widetilde W$, the slope of $\widetilde \alpha_k = \widetilde\mu + 2k\widetilde\lambda$ is identified with $\widetilde\tau + 2k\widetilde\sigma = (q-2k)\widetilde\gamma + (2k+1-q)\widetilde\varphi$.  Then, $\tW(\alpha_k)$ has Seifert invariants 
\[
M\left(-1,\frac{(q-1)/2}{q},\frac{(q-1)/2}{q},\frac{1}{q-2k}\right).
\]
Since $\frac{(q-1)/2}q < \frac12$, and $0<\frac1{q-2k} < \frac12$ if $k\leq 0$, $\tW(\widetilde\alpha_k)$ admits a horizontal foliation by \eqref{condition:-1} of Theorem~\ref{thm:brwcondition}.

Second, we have $\ttau = q \tgamma + (-1-q) \tphi$ and $\tsigma = \tgamma - \tphi$.  Then, $\tW(\alpha_k)$ has Seifert invariants 
\[
M\left(-1,\frac{(q-1)/2}{q},\frac{(q-1)/2}{q},\frac{-1}{2k+q}\right).
\]
Since $\frac{(q-1)/2}q < \frac12$, and $0<\frac{-1}{2k+q} < \frac12$ if $k \ll 0$, $\tW(\widetilde\alpha_k)$ admits a horizontal foliation by \eqref{condition:-1} of Theorem~\ref{thm:brwcondition}.  
\end{proof}

We next consider the case $\gcd(n,q) \ne 1$, i.e. $\widetilde J$ is not connected.

\begin{proposition}\label{prop:234} 
Let $K$ be a non-trivial knot in $S^3$. 
Then the manifolds $\Sigma_2 (C_{3,4}(K))$, $\Sigma_4(C_{3,2}(K))$, and $\Sigma_3 (C_{2,3}(K))$ are excellent.
\end{proposition}

\begin{proof} 
We use the same arguments as in Proposition~\ref{prop:exceptional}.  As before, it suffices to show that $\widetilde{W}(\widetilde\alpha_k,\ldots,\widetilde\alpha_k)$ admits a horizontal foliation.  Recall that $\ttau = \omega\tgamma + \eta \tphi$, where $\omega = \frac{nq}{\gcd(n,q)\gcd(n,p)}$, and we solve for the possible values of $\eta$ using $\ttau \cdot \tsigma = 1$ .  \\

\noindent {\bf Case 1:} $(n,p,q) = (2,3,4)$. \\
In this case $\widetilde J$ has two components, each mapping homeomorphically to $J$.  By \cite[Theorem~1]{NR}, $\Sigma_2 (T_{3,4}) = M(\frac12, -\frac13, -\frac13)$.   
Since $J$ has multiplicity~3, each component of $\widetilde J$ is a fiber of multiplicity~3.  
Thus, $\widetilde\sigma_i = \pm(-3\widetilde\gamma_i + \widetilde\varphi_i)$. In this case, there is only one choice for $\eta$, and we find $\tsigma_i = -3\tgamma_i + \tphi_i$ and $\ttau_i = 4\tgamma_i - \widetilde\varphi_i$, for $i = 0,1$.

We have $\Sigma_2 (C_{3,4}(K)) \cong X_0 \cup X_1 \cup \widetilde W$, where  $X_i$ is a copy of $X$, glued along the two boundary components of $\widetilde W$. 
On $\partial X_i$ the slope $\alpha_k = \mu_i +k\lambda_i$ is identified with $\widetilde\tau_i + k\widetilde\sigma_i = (4-3k)\widetilde\gamma_i + (k-1)\widetilde\varphi_i$. 
Hence $\widetilde W(\alpha_k, \alpha_k)$ is the Seifert fibered space $M(\frac12, \frac{k-1}{4-3k}, \frac{k-1}{4-3k})  = -M( -1,\frac12, \frac{k-1}{3k-4}, \frac{k-1}{3k-4})$.
Since $0<\frac{k-1}{3k-4} < \frac13$ if $k\le 0$, $\widetilde W(\alpha_k,\alpha_k)$ has a horizontal foliation by Condition \eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}.  \\

\noindent {\bf Case 2:} $(n,p,q) = (4,3,2)$. \\
By \cite[Theorem 1]{NR}, $\Sigma_4 (T_{3,2}) = M(-\frac12, -\frac13, -\frac13,\frac11)$. The curve $J$ has multiplicity~3, $\widetilde J$ has two components, each mapping to $J$ by a covering map of degree~2, and each being a fiber of multiplicity~3. 
So, $\widetilde\sigma_i = \pm(-3\widetilde\gamma_i + \widetilde \varphi_i)$.  There is only one choice for $\eta$, and we see $\widetilde\tau_i = 4\widetilde\gamma_i - \widetilde\varphi_i$ and $\tsigma_i = -3\tgamma_i + \tphi_i$.  

The manifold $\Sigma_4 (C_{3,2}(K)) \cong X^{(2)}_0 \cup X^{(2)}_1 \cup \widetilde W$, where  each $X^{(2)}_i$ is a copy of $X^{(2)}$. 
The slope $\widetilde\alpha_k = \widetilde\mu_i + 2k\widetilde\lambda_i$ on $\partial X_i^{(2)}$ is identified with $\widetilde\tau_i + 2k\widetilde\sigma_i = (4-6k)\widetilde\gamma_i + (2k-1)\widetilde\varphi_i $. 
So $\widetilde W (\widetilde\alpha_k,\widetilde\alpha_k)$ is the Seifert manifold $M(1,-\frac12, \frac{2k-1}{4-6k},\frac{2k-1}{4-6k}) = -M(-1,\frac12,\frac{2k-1}{6k-4},\frac{2k-1}{6k-4})$.   
For $k\le 0$, we have $\frac{2k-1}{6k-4} < \frac13$, so the result follows from Condition~\eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition}. \\

\noindent {\bf Case 3:} $(n,p,q) = (3,2,3)$. \\
By \cite[Theorem~1]{NR}, $\Sigma_3 (T_{2,3}) = M(-\frac12, -\frac12, -\frac12, \frac11)$.  In this case, 
$J$ has multiplicity~2, $\widetilde J$ has three components, and each component of $\widetilde J$ has multiplicity~2. 
Therefore, $\widetilde\sigma_i = \pm (-2\widetilde\gamma_i + \widetilde\varphi_i)$.  There are two cases.  

First, $\eta = -1$.  We have $\ttau_i = 3 \tgamma_i - \tphi_i$, $\tsigma_i = -2\tgamma_i + \tphi_i$.  We have $\Sigma_3 (C_{2,3}(K)) \cong X_0\cup X_1 \cup X_2 \cup \widetilde W$, where each $X_i$ is a copy of $X$.  The gluing of the boundaries identifies $\alpha_k = \mu_i + k\lambda_i$ with $\widetilde\tau_i + k\widetilde\sigma_i = (3-2k)\widetilde\gamma_i + (k-1)\widetilde\varphi_i$.  Therefore $\widetilde W (\alpha_k,\alpha_k,\alpha_k)$ is the Seifert fibered space $M(1, \frac{k-1}{3-2k}, \frac{k-1}{3-2k}, \frac{k-1}{3-2k}) = -M(-1,\frac{k-1}{2k-3},\frac{k-1}{2k-3},\frac{k-1}{2k-3})$.   
Since $0<\frac{k-1}{2k-3} < \frac12$ if $k\leq 0$, Condition \eqref{condition:-(n-1)} of Theorem~\ref{thm:brwcondition} implies that $\widetilde W$ has a horizontal foliation.


Second, $\eta = -2$.  We have $\ttau_i = 3 \tgamma_i - 2\tphi_i$, $\tsigma_i = 2\tgamma_i - \tphi_i$.  Therefore $\widetilde W (\alpha_k,\alpha_k,\alpha_k)$ is the Seifert fibered space $-M(-1, \frac{k+2}{2k+3}, \frac{k+2}{2k+3}, \frac{k+2}{2k+3})$. 
Since $0<\frac{k+2}{2k+3} < \frac12$ if $k \ll 0$, Condition \eqref{condition:-1} of Theorem~\ref{thm:brwcondition} implies that $\widetilde W$ has a horizontal foliation.
\end{proof}


The following completes the proof of Theorem~\ref{thm:cables}.

\begin{proposition}\label{prop:1q}
Let $K$ be a non-trivial knot in $S^3$. 
Unless $n=q=2$, $\Sigma_n (C_{1,q}(K))$ is excellent.
\end{proposition}

\begin{proof}
Let $d = \gcd(n,q)$ and let $r = \frac{n}{d}$ and $s = \frac{q}{d}$.  Note that $\gcd(r,s) = 1$.  From the discussion immediately preceding Proposition~\ref{prop:generic}, we have that $\Sigma_n(C_{1,q}(K)) = \widetilde{W} \cup \coprod^d_{i=1} X^{(r)}_i $, where 
\[
\widetilde{W} = \Sigma_n(T_{1,q}) - \coprod^d_{i=1} int(\widetilde{V}_i) = S^3 - int(N(\widetilde{J})).
\]
To identify $\widetilde{J} \subset S^3$, it is convenient to note that there is an isotopy of $S^3$ that interchanges the components of $C$ and $J$.  With this picture in mind, it is then clear that $\widetilde{J}$ is the $(n,q)$-torus link $T_{n,q} = T_{dr,ds}$.  Let $\widetilde{J}_1,\ldots,\widetilde{J}_d$ denote the components of $\widetilde{J}$.  

As before, let $\widetilde{\sigma}_i$ be the meridian of $\widetilde{V}_i$ and $\widetilde \tau_i \subset \partial \widetilde{V}_i$, the lift of $\tau$.  Let $\widetilde{\rho}_i \subset \partial \widetilde{V}_i$ be the 0-framed longitude of $\widetilde{J}_i \subset \Sigma_n(T_{1,q}) \cong S^3$.  Since $\widetilde \tau_i \cdot \widetilde{\sigma}_i = 1$ on $\partial \widetilde{W}$, we have $\widetilde \tau_i = \widetilde \rho_i + c \widetilde \sigma_i$ for some $c \in \mathbb{Z}$.  Let $\alpha_k$ be the slope $\mu + k\lambda$ on $\partial X$, and $\widetilde{\alpha}_k = \widetilde \mu_i + rk\widetilde \lambda_i$ the corresponding slope on $\partial X^{(r)}_i$.  Under the gluing homeomorphism $\partial X^{(r)}_i \to \partial \widetilde{V}_i \subset \partial \widetilde W$, we have that $\widetilde{\alpha}_k$ is mapped to $\widetilde \tau_i + rk \widetilde \sigma_i = \widetilde \rho_i + (c + rk) \widetilde \sigma_i$.  By Proposition~\ref{prop:negtorus}, for $k$ sufficiently negative, the multislope $(\widetilde{\alpha}_k,\ldots,\widetilde{\alpha}_k)$ is excellent for $\widetilde{W}$, unless $n = q = 2$.  By Lemma~\ref{lem:cover1/n}, we have that for $k \ll 0$, $\widetilde{\alpha}_k$ is an excellent slope for for each $X^{(r)}_i$.  The result again follows from Lemma~\ref{lem:excellentgluing}.     
\end{proof}


\subsection{Cyclic branched covers of $(p,2)$-cables}
We now prove Theorems~\ref{thm:conditional} and \ref{thm:cablefail}.  Let $p \geq 1$ be odd.  In this case, the link $C \cup J$ from the discussion just before Proposition~\ref{prop:generic} is illustrated in Figures~\ref{fig:cables2fig1} and ~\ref{fig:cables2fig2}, which show the cases $p = 5$ and $p = 1$ respectively.  For the special case of $p = 1$, we relabel the components as $C' \cup J'$, with associated meridian-longitude pair $\mu', \lambda'$.  

\begin{figure}
\labellist 
\large
\pinlabel  $C$ at 98 167
\pinlabel  $J$ at 130 77
\endlabellist
\includegraphics[scale=.7]{fig1}
\caption{The union of a regular fiber, $C$, and an exceptional fiber, $J$, of multiplicity $p = 5$, in a Seifert fibration of $S^3$.}\label{fig:cables2fig1}
\end{figure}
\begin{figure}
\labellist 
\large
\pinlabel  $C$ at 81 110
\pinlabel  $J$ at 114 42
\endlabellist
\includegraphics[scale=.8]{fig2}
\caption{The union of a regular fiber, $C$, and an exceptional fiber, $J$, of multiplicity $p = 1$ in a Seifert fibration of $S^3$.}\label{fig:cables2fig2}
\end{figure}

Now, for any $p$, the exterior of $C \cup J$ is in fact homeomorphic to $C' \cup J'$.  One way to see this is as follows.  The exterior of $C \cup J$ is the exterior of a regular fiber in the $\frac{2 \pi p}{2}$ Seifert fibered solid torus.  Therefore, applying $r = \frac{p-1}{2}$ negative meridional Dehn twists to this solid torus produces a $\frac{2 \pi 1}{2}$ Seifert fibered solid torus; this is the exterior of $C' \cup J'$.  In other words, there is a homeomorphism $h: \overline{W - N(C)} \to \overline{W' - N(C')}$ such that $h(\tau) = \tau'$ and $h(\sigma) = \sigma' - r \tau'$.  

We may interchange the components of $C' \cup J'$ to obtain the link in Figure~\ref{fig:cables2fig3}.  
\begin{figure}
\includegraphics[scale=.8]{fig3}
\labellist 
\large
\pinlabel  $J'$ at 90 338
\pinlabel  $C'$ at 135 250
\endlabellist
\caption{The result of interchanging the components of $C' \cup J'$.}\label{fig:cables2fig3}
\end{figure}
From this, we can see that $\widetilde{J}' \subset \Sigma_2(T_{1,2}) \cong S^3$ is the Hopf link (see Figure~\ref{fig:cables2fighopf}).  

\begin{figure}
\includegraphics[scale=.65]{fighopf}
\labellist 
\large
\pinlabel  $J$ at 85 335
\pinlabel  $C$ at 135 250
\endlabellist
\caption{The lift, $\widetilde{J}'$, of $J'$ in $\Sigma_2(T_{1,2})$.}\label{fig:cables2fighopf}
\end{figure}
Let the components of $\widetilde{J}'$ be $J'_0$ and $J'_1$, and let the corresponding lifts of $V'$ be $V'_0$ and $V'_1$.  Then $\widetilde W' \cong \mathbb{T}^2 \times I$, where $\partial V'_i = \mathbb{T}^2 \times \{i\}$, for $i = 0,1$.  Consequently, we have the corresponding decomposition $\Sigma_2(T_{p,2}) = V_0 \cup V_1 \cup \widetilde{W}$.  We would like to determine the gluing.  If $\alpha$ is a slope on $\partial V$ (respectively $\partial V'$), let $\alpha_i$ (respectively $\alpha'_i$) be the corresponding slope on $\partial V_i$ (respectively $\partial V'_i$), for $i = 0,1$.  Note that the homeomorphism $h$ lifts to a homeomorphism $\widetilde{h}:\widetilde W \to \widetilde W'$ such that $\widetilde h(\tau_i) = \tau_i$ and $\widetilde h(\sigma_i) = \sigma'_i - r\tau'_i$, for $i = 0,1$.  

Let $\beta'$ be the blackboard framed longitude of $J'$ in Figure~\ref{fig:cables2fig3}.  Then $\beta' = \tau' + \sigma'$, and $\beta'_i = \tau'_i + \sigma'_i$ is the 0-framed longitude of $J'_i$.  

Recall that $\Sigma_2(C_{p,2}(K)) \cong X_0 \cup X_1 \cup \widetilde{W}$, where $X_i$ is the copy of the exterior $X$ of $K$, for $i = 0,1$, and the gluing homeomorphism $\partial X_i \to \partial V_i$ takes $\lambda_i$ to $\sigma_i$ and $\mu_i$ to $\tau_i$, for $i = 0,1$.  Using the homeomorphism $\widetilde{h}$, $\Sigma_2(C_{p,2}(K)) \cong X_0 \cup X_1 \cup \widetilde{W}'$, where the gluing homeomorphism $\partial X_i \to \partial V'_i = \mathbb{T}^2 \times \{i\}$ now takes $\lambda_i$ to 
\[
\sigma'_i - r\tau'_i = \sigma'_i - r(\beta'_i - \sigma'_i) = (r+1) \sigma'_i - r\beta'_i,
\]
and takes $\mu_i$ to $\tau'_i = - \sigma'_i + \beta'_i$, for $i = 0,1$.  


Thus, with respect to the ordered bases $\lambda_i, \mu_i$ and $\sigma'_i,\beta'_i$, this gluing homeomorphism is given by the matrix $A = \begin{pmatrix} r+1 & -1 \\ -r & 1 \end{pmatrix}$.  Since $\widetilde{W}' \cong \mathbb{T}^2 \times I$, with $\sigma'_0, \beta'_0$ homotopic to $\beta'_1, \sigma'_1$, respectively, it follows that $\Sigma_2(C_{p,2}(K)) \cong X_0 \cup X_1$, glued by the homeomorphism $f: \partial X_0 \to \partial X_1$ that is given with respect to the ordered bases $\lambda_0,\mu_0$ and $\lambda_1,\mu_1$, by the matrix $A^{-1}BA$, where $B = \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$, i.e. by $\begin{pmatrix} 1 & 0 \\ 2r+1 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ p & -1 \end{pmatrix}$.  In other words, we have 
\begin{equation}\label{eqn:cp2}
\Sigma_2(C_{p,2}(K))  = X_0 \cup_f X_1, \; \; f(a \mu_0 + b \lambda_0) =  (pb-a)\mu_1 + b \lambda_1.  
\end{equation}
Observe that under this identification, a slope $\frac{a}{b}$ on $\partial X_0$ is sent to the slope $p - \frac{a}{b}$ on $\partial X_1$.  


\begin{proof}[Proof of Theorem~\ref{thm:conditional}]
From \eqref{eqn:cp2}, we have $\Sigma_2(C_{1,2}(K)) = X_0 \cup X_1$, where the slope $\frac{a}{b}$ is identified with $1-\frac{a}{b}$.  The slopes $\frac{1}{2}$ on $\partial X_0$ and $\partial X_1$ are therefore identified.  If $\frac{1}{2}$ is a CTF slope for $K$, then $\frac{1}{2}$ is an excellent slope by Lemma~\ref{lem:toroidalzs3}.  Thus, $\Sigma_2(C_{1,2}(K))$ is excellent by Lemma~\ref{lem:excellentgluing}.
\end{proof}


\begin{proof}[Proof of Theorem~\ref{thm:cablefail}]
By \eqref{eqn:cp2}, $\Sigma_2(C_{p,2}(T_{2,3})) \cong X_0 \cup_\partial X_1$, where $X_0, X_1$ are copies of the exterior of $T_{2,3}$ and the gluing homeomorphism $\partial X_0 \to \partial X_1$ takes a slope $\frac{a}{b}$ to $p-\frac{a}{b}$.  

By \cite[p.10]{RobertsPunctured1} and Theorem~\ref{thm:brwcondition}, the set of CTF slopes for $T_{2,3}$ is exactly $(-\infty,1)$.  First, let $p  = 1$.  Since $\frac{1}{2}$ is a CTF slope for $T_{2,3}$, we have that $\Sigma_2(C_{1,2}(T_{2,3}))$ is excellent by Theorem~\ref{thm:conditional}.  Next, we consider $\Sigma_2(C_{p,2}(T_{2,3}))$,  for $p \leq -1$, or equivalently, $-\Sigma_2(C_{p,2}(T_{-2,3}))$ for $p \geq 1$.   We have that the set of CTF slopes for $T_{-2,3}$ is $(-1,\infty)$.  In this case, we have that $\frac{1}{2}$ is a CTF slope for $T_{-2,3}$ which is identified with $p - \frac{1}{2} \in (-1,\infty)$, a CTF slope for $T_{-2,3}$.  Since the exterior of $T_{-2,3}$ is Seifert fibered, we have that $p - \frac{1}{2}$ is an excellent slope by Lemma~\ref{lem:seifertboundary}.  Therefore, $\Sigma_2(C_{p,2}(T_{-2,3}))$ is excellent for $p \geq 1$ by Lemma~\ref{lem:excellentgluing}.  


Finally, we would like to show that $\Sigma_2(C_{p,2}(T_{2,3}))$ is not excellent for $p \geq 3$.  We do this by showing that this manifold does not admit a co-orientable taut foliation; further, by \cite[Theorem 1.1]{BC}, since $\Sigma_2(C_{p,2}(T_{2,3})$ is a graph-manifold, $\pi_1(\Sigma_2(C_{p,2}(T_{2,3})))$ will not be left-orderable.  By \cite[Theorem 1.7]{BC}, $\Sigma_2(C_{p,2}(T_{2,3}))$ will not admit a co-orientable taut foliation if we can show that no foliation-detected slope on $\partial X_0$ is identified with a foliation-detected slope on $\partial X_1$.  For the definition of a foliation-detected slope, see \cite[Definition 6.1]{BC}.  It follows from the definition that any CTF slope on $X_i$ is a foliation-detected slope.  By the computation of the gluing map between $\partial X_0$ and $\partial X_1$, it suffices to show that the set of foliation-detected slopes on $T_{2,3}$ is the interval $(-\infty,1]$.  As discussed above, the set of CTF slopes is precisely $(-\infty,1)$.  Since the set of foliation-detected slopes on a Seifert manifold with connected boundary and base orbifold $S^2$ is the closure of the set of CTF slopes by \cite[Theorem 1.6, Proposition A.4(4)(b), Corollary A.7]{BC}, we have the desired result.       

\end{proof}
{"config": "arxiv", "file": "1406.6718/cables.tex"}