  While the topological types of {normal} surface singularities with homology
sphere link have been classified, forming a rich class, until recently little
was known about the possible analytic structures. We proved in [Geom. Topol.
9(2005) 699-755] that many of them can be realized as complete intersection
singularities of "splice type", generalizing Brieskorn type.
  We show that a normal singularity with homology sphere link is of splice type
if and only if some naturally occurring knots in the singularity link are
themselves links of hypersurface sections of the singular point.
  The Casson Invariant Conjecture (CIC) asserts that for a complete
intersection surface singularity whose link is an integral homology sphere, the
Casson invariant of that link is one-eighth the signature of the Milnor fiber.
In this paper we prove CIC for a large class of splice type singularities.
  The CIC suggests (and is motivated by the idea) that the Milnor fiber of a
complete intersection singularity with homology sphere link Sigma should be a
4-manifold canonically associated to Sigma. We propose, and verify in a
non-trivial case, a stronger conjecture than the CIC for splice type complete
intersections: a precise topological description of the Milnor fiber. We also
point out recent counterexamples to some overly optimistic earlier conjectures
in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann.
326(2003) 75--93].
