state it, let $\alpha\in \HFa(Y,\spinc)$ be a non-vanishing Floer homology class, and let $\iota_m: \FiltY(K,m)\rightarrow \CFa(Y,\spinc)$ be the inclusion map.

\[defn:tau\_alpha\] $$\tau_{\alpha}(Y,K)=\mathrm{min}\{m\in\Z|\ \alpha \subset \mathrm{Im} \ (\iota_m)_*\},$$

where $(\iota_m)_*$ is the map induced on homology.

Proof of Theorem \[thm:filt\]
-----------------------------

With the above notation, we will prove the following more general version of Theorem \[thm:filt\]:

\[thm:filtY\] Let $K\subset Y$ be a knot and let $\spinc\in\SpinC(Y)$. Pick any $M\in \Z$. Then there exists a constant $N(M)>0$ so that $\forall \ n>N(M)$, the following holds for each $j> M$: $$H_*(\FiltY(K_{p,pn+1},pj+\frac{(pn)(p-1)}{2}-1))\cong H_*(\FiltY(K,j-1)).$$ Furthermore, $$\begin{array}{ll}
    H_*(\FiltY(K_{p,pn+1},pj+\frac{(pn)(p-1)}{2}-i))\cong &  \\ 
H_*(\FiltY(K_{p,pn+1},pj+\frac{(pn)(p-1)}{2}-i-1)) & \forall \ i=2,\ldots,p-1. \\ 
\end{array}$$ In particular, $$\tau_\alpha(Y,K_{p,pn+1})=\left\{\begin{array}{ll}
        p \tau_\alpha(Y,K)+\frac{(pn)(p-1)}{2}+p-1 & {\text{or}} \\
      p \tau_\alpha(Y,K)+\frac{(pn)(p-1)}{2}. & \\ 
\end{array}
\right.$$

In the above, the filtration for $K_{p,pn+1}$ is defined using a Seifert surface $F_{p,pn+1}$ satisfying $[F_{p,pn+1}]=p[F]$, where $F$ is the Seifert surface used to define the filtration for $K$.

The key tool in proving the theorem is

\[lemma:hd\]Lemma 2.2 of [@Cabling]Let $$H=(\Sigma,\{\alpha_1,\ldots,\alpha_g\},\{\beta_1,\ldots,\beta_{g-1},\mu\},z,w),$$ be a Heegaard diagram for a knot K, where $\mu$ is a meridian for $K$. Then $$H(p,n)= (\Sigma,\{\alpha_1,\ldots,\alpha_g\},\{\beta_1,\ldots,\beta_{g-1},\tilde{\beta}\},z',w),$$

is a Heegaard diagram for $K_{p,pn+1}$. $H$ and $H(p,n)$ differ only in the final curve $\tilde{\beta}$. Here, $\tilde{\beta}$ is obtained by winding $\mu$ around an 