$K$. It follows from the adjunction inequality for knot Floer homology that

$$\begin{array}{ll} H_*(\FiltY(K,j))=0  & \forall j<-g(K). \end{array}$$

The second part of the theorem now follows from the definition of $\tau_\alpha$.                            $\square$

Examining the Heegaard diagram $H(p,n)$ when $n<0$, we are lead to:

\[thm:filtYneg\] 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 $\forall \ n>N$, 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, $$\begin{array}{ll}
    H_*(\FiltY(K_{p,-pn+1},pj-\frac{(pn)(p-1)}{2}+i))\cong &  \\ 
H_*(\FiltY(K_{p,-pn+1},pj-\frac{(pn)(p-1)}{2}+i+1)) & \forall \ i=1,\ldots,p-2. \\ 
\end{array}$$ In particular, $$\tau_\alpha(Y,K_{p,-pn+1})=\left\{\begin{array}{ll}
        p \tau_\alpha(Y,K)-\frac{(pn)(p-1)}{2}+p-1 & {\text{or}} \\
      p \tau_\alpha(Y,K)-\frac{(pn)(p-1)}{2}. & \\ 
\end{array}
\right.$$

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

We now turn to the proof of Theorem \[thm:tau\]. We focus first on proving the inequality: $$\label{eq:tauinequality} p \tau(K)+ \frac{(pn)(p-1)}{2}\ \  \le\tau(K_{p,pn+1})\le \ \ 
     p \tau(K)+\frac{(pn)(p-1)}{2} +p-1.$$

This will follow rather quickly from Theorems \[thm:filtY\] and \[thm:filtYneg\], together with a crossing change inequality satisfied by $\tau$ to interpolate between the cases $n>0$ and $n<0$. Recall that two knots $K_+, K_- \subset S^3$ are said to differ by a crossing change if there exists an embedded three-ball $B^3\subset S^3$, outside of which the knots agree,$$(S^3,K_+)\backslash B^3 \cong (S^3,K_-)\backslash B^3,$$ and such that the 