p \tau(K)+\frac{(pn)(p-1)}{2}+p-1 & {\text{or}} \\
      p \tau(K)+\frac{(pn)(p-1)}{2}. & \\ 
\end{array}
\right.$$

The theorem has an analogue for $n<0$ (stated in Section \[sec:proof\]) which we use with the above to prove:

\[thm:tau\]

Let $K\subset S^3$ be a non-trivial knot, then the following inequality holds for all $n$, $$p \tau(K)+ \frac{(pn)(p-1)}{2}\ \  \le\tau(K_{p,pn+1})\le \ \ 
        p\tau(K)+\frac{(pn)(p-1)}{2} +p-1.$$

In the special case when $K$ satisfies $\tau(K)=g(K)$ we have the equality, $$\tau(K_{p,pn+1}) =  p \tau(K)+\frac{(pn)(p-1)}{2},$$ whereas when $\tau(K)=-g(K)$ we have $$\tau(K_{p,pn+1}) =  p \tau(K)+\frac{(pn)(p-1)}{2} + p-1.$$

Here, and throughout, $g(K)$ denotes the Seifert genus of $K$. We emphasize that while Theorem \[thm:filt\] requires the cabling parameter $n$ to be sufficiently large, there is no restriction on $n$ in the statement of Theorem \[thm:tau\].

Recently, we generalized $\tau(K)$ to a sequence of invariants of a knot $K\subset Y$ in an arbitrary $3$-manifold [@tbbounds]. For each non-vanishing Floer homology class, $\alpha\in \HFa(Y,\spinc)$ we obtain an integer, $\tau_\alpha(Y,K)$. Theorem \[thm:filt\] is a special case of Theorem \[thm:filtY\], found in the next section. This latter theorem holds for knots in arbitrary manifolds and we use this more general result to obtain information for each $\tau_\alpha(Y,K)$.

Of particular interest is the case when $\alpha$ is the  contact invariant, $c(\xi)\in 