  In Theorem 1.2 of the paper math.GT/0002110 the author claimed to have proved
that all transversal knots whose topological knot type is that of an iterated
torus knot (we call them cable knots) are transversally simple. That theorem is
false, and the Erratum math.GT/0610565 identifies the gap. The purpose of this
paper is to explore the situation more deeply, in order to pinpoint exactly
which cable knots are {\it not} transversally simple. The class is subtle and
interesting. We will recover the strength of the main theorem in
math.GT/0002110, in the sense that we will be able to prove a strong theorem
about cable knots, but the theorem itself is more subtle than Theorem 1.2 of
math.GT/0002110.
  In particular, we give a geometric realization of the Honda-Etnyre transverse
(2,3)-cable of the (2,3)-torus knot example (Appendix joint with H. Matsuda).
(See math.SG/0306330)
