Minimal Page-Genus of Milnor Open Books on Links of
Rational Surface Singularities
Selma Altınok and Mohan Bhupal
In light of the fundamental theorem of Giroux  relating contact structures on
a 3-manifold M to open book decompositions of M, it is interesting to ask what the
minimal page-genus of an open book supporting a given contact structure on a 3-
manifold is. In general, not much is known in answer to this question. However, for
some special classes of contact structures there are some results (see, for example,
, , , , ). Here we consider the class of Milnor ﬁllable contact structures
on links of rational surface singularities. Since the associated Milnor open books
all support the given contact structure (see ), one may consider minimizing the
page-genus over this smaller class of open books. Here we prove the following result.
Theorem 1.1. Let (X, x) be a germ of a normal complex analytic surface
having a rational singularity at x. The page-genus of the associated Milnor open
books OB(f) is minimized when f is taken to be the restriction of a generic linear
form on CN to (X, x) for some/any local embedding of (X, x) in (CN , 0).
A related problem is to try to minimize the sum of the page-genus plus the
number of binding components. Here we have the following result.
Theorem 1.2. With the same hypotheses as Theorem 1.1, the page-genus plus
the number of binding components of OB(f) is minimized when f is taken to be the
restriction of a generic linear form on CN to (X, x) for some/any local embedding
of (X, x) in
2.1. Contact structures and open books. All our manifolds are assumed
to be oriented.
A contact structure on a 3-manifold M is a 2-plane ﬁeld ξ ⊂ T M that can be
written as the kernel of a 1-form α ∈
α ∧ dα 0.
2000 Mathematics Subject Classiﬁcation. 57R17, 53D10,32S25, 32S55.
Key words and phrases. Contact structure, Milnor open book, minimal page-genus, rational
singularity, Tjurina component, non-Tjurina component.
Volume 475, 2008
c 0000 (copyright holder) c 2008 American Mathematical Society