We say that a contact manifold (M, ξ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X , x). In this article we prove that any 3-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate with any holomorphic function f : (X , x) → (C, 0), with isolated singularity at x (and any euclidian rug function ρ), an open book decomposition of M , and we verify that all these open books carry the contact structure ξ of (M, ξ) -generalizing results of Milnor and Giroux.

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ > 1 and λ λ−1 .

The oriented link of the cyclic quotient singularity 𝒳p, q is orientation‐preserving diffeomorphic to the lens space L(p, q) and carries the standard contact structure ξst. Lisca classified the Stein fillings of (L(p, q), ξst) up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the Milnor fibres associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of 𝒳p, q. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibres given by de Jong and van Straten, we also canonically identify these fibres with Lisca's fillings. Using these and a newly introduced additional structure (the order) associated with lens spaces, we prove that the above Milnor fibres are pairwise non‐diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.