\circ \phi_{p,pn+1}([K]) = \tau(K_{p,pn+1}).$$

Theorem \[thm:tau\] shows that $\ctau{n}$ is a piecewise linear function of $n$. Indeed, the graph of $\ctau{n}$ lies entirely on the lines of slope $\frac{(p-1)(p)}{2}$ whose $y$ intercepts range between $p\tau(K)$ and $p\tau(K)+p-1$. Moreover, a crossing change inequality for $\tau$ (Equation ) shows that $\ctau{n}$ is monotonically decreasing. This indicates that $\ctau{n}$ has a finite set of discontinuities: $$J_\tau^p(K)= \{ n\in \Z | \ \ctau{n} \ne \ctau{n+1}-\frac{(p-1)(p)}{2} \}.$$ The cardinality of this “jumping locus" is at most $p-1$ and may be zero. For instance, Theorem \[thm:tau\] indicates $J_p(K)=\emptyset$ if $\tau(K)=\pm g(K)\ne 0$. On the other hand, for the unknot we have $J_p(\text{unknot})=\{-1\}$.

One should compare this with recent work of Van Cott [@VanCott2008], which uses formal properties of $\tau$ to reprove the fact that $\ctau{n}$ is bounded between two lines of slope $\frac{(p-1)p}{2}$. Her results extend to $(p,q)$ cables, but are unable to recover the possible $y$ intercepts of the lines which bound the graph of $\ctau{n}$, showing only that they differ by $p-1$.

In light of this, one might hope that formal properties of $\tau$ could be pushed further to reprove Theorem \[thm:tau\] without having to understand the Floer chain complexes. This seems unlikely, due 