Contemporary Mathematics

Minimal Page-Genus of Milnor Open Books on Links of

Rational Surface Singularities

Selma Altınok and Mohan Bhupal

1. Introduction

In light of the fundamental theorem of Giroux [9] 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,

[6], [15], [7], [5], [2]). 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 [4]), 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

(CN

, 0).

2. Preliminaries

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 ∈

Ω1(M)

such that

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.

1

Contemporary Mathematics

Volume 475, 2008

1

c 0000 (copyright holder) c 2008 American Mathematical Society

http://dx.doi.org/10.1090/conm/475/09272

Minimal Page-Genus of Milnor Open Books on Links of

Rational Surface Singularities

Selma Altınok and Mohan Bhupal

1. Introduction

In light of the fundamental theorem of Giroux [9] 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,

[6], [15], [7], [5], [2]). 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 [4]), 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

(CN

, 0).

2. Preliminaries

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 ∈

Ω1(M)

such that

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.

1

Contemporary Mathematics

Volume 475, 2008

1

c 0000 (copyright holder) c 2008 American Mathematical Society

http://dx.doi.org/10.1090/conm/475/09272