positive surgery on $K_{p,pn+1}$ yields a lens space or, more generally, an L-space. Then $n\ge 0$ and $\tau(K)=g(K)$.

As counterpoint to this obstruction, we also have the following existence theorem:

\[thm:lspace\] Let $K\subset S^3$ be a non-trivial knot which admits a positive $L$-space space surgery. Then $K_{p,q}$ admits positive L-space surgeries whenever $q\ge p(2g(K)-1)$,

Note the above result does not require $q$ of the form $q=pn+1$. Indeed, it is proved somewhat differently from Theorem \[thm:filt\], using a standard cut-and-paste topological argument together with known properties of the  invariants.

Theorem \[thm:lspace\] is interesting in light of a paucity of examples. The theorem is the first general construction of L-space knots outside of the double primitive knots [@Berge]. Indeed, any sufficiently positive iterated cable of a knot which has an actual lens space surgery will itself have L-space surgeries.

As noted above, L-space knots have the property that the $\Z\oplus\Z$ filtered chain homotopy type of the knot’s filtration of $\CF^\infty(S^3)$ is determined by the Alexander polynomial [@Lens]. Since the Alexander polynomial of a cable knot is determined by Formula , Theorem \[thm:lspace\] provides an efficient method for calculating the Floer homology of a large class of cable knots. For instance, $+5$ surgery 