the Thurston norm.

The purpose of this paper is to extend our knowledge of the filtered chain homotopy type of $\Filt(K_{p,pn\pm1})$ beyond the level of its associated graded object. A primary motivation for this extension comes from the relationship between the filtered chain homotopy type of $\Filt(K)$ and the smooth four-ball genus of $K$, $g_4(K)$.  [@FourBall] and Rasmussen [@RasThesis] define the following numerical invariant of a knot, $K\subset S^3$: $$\tau(K)=\mathrm{min}\{i\in\Z|H_*(\Filt(K,i))\longrightarrow \HFa(S^3)\ \mathrm{is \ non\-trivial}\}.$$

It is shown that this invariant provides a lower bound for the four-ball genus, $$|\tau(K)|\le g_4(K),$$ and that $\tau$ provides a homomorphism from the the smooth concordance group of knots, $\mathcal{C}$, to $\Z$. Moreover, the above inequality is sharp for torus knots, providing a new proof of Milnor’s famous conjecture [@Milnor1968] on the four-genera and unknotting numbers of this family.

The main result of this paper is the following theorem which, for simplicity, we state for knots in the three-sphere.

\[thm:filt\] Let $K\subset S^3$ be a knot. Pick any $M\in \Z$. Then there exists a constant $N>0$ so that $\forall \ n>N$, the following holds for each $j> M$: $$H_*(\Filt(K_{p,pn+1},pj+\frac{(pn)(p-1)}{2}-1))\cong H_*(\Filt(K,j-1)).$$ Furthermore, $$\begin{array}{ll}
    H_*(\Filt(K_{p,pn+1},pj+\frac{(pn)(p-1)}{2}-i))\cong &  \\ 
H_*(\Filt(K_{p,pn+1},pj+\frac{(pn)(p-1)}{2}-i-1)) & \forall \ i=2,\ldots,p-1. \\ 
\end{array}$$ In particular, $$\tau(K_{p,pn+1})=\left\{\begin{array}{ll}
        