Abstract
In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate whether the Milnor genus is equal to the support genus for links of quotient surface singularities. We show that for many types of the quotient surface singularities the Milnor genus is equal to the support genus. In the remaining cases we are able to find a small upper bound for the support genus.
Highlights
The purpose of this paper is to construct the Milnor open book decompositions of the links of quotient surface singularities supporting the unique Milnor fillable contact structure
If we restrict ourself to quotient surface singularities, the question whether the Milnor genus is equal to the support genus for the canonical contact structure is still unknown
For most cases of the quotient surface singularities, we provide planar Milnor open books, so that for these types the Milnor genus is equal to the support genus
Summary
The purpose of this paper is to construct the Milnor open book decompositions of the links of quotient surface singularities supporting the unique Milnor fillable contact structure. For most cases of the quotient surface singularities, we provide planar Milnor open books, so that for these types the Milnor genus is equal to the support genus. The unique Milnor fillable contact structure on the link of the quotient surface singularities has support genus one for each singularities of the following types:. Open book decomposition, contact structure, Milnor genus, support genus. For icosahedral singularity of part (i) where b = 2, it was shown in [E] that contact structure cannot be supported by a planar open book decomposition. It was shown in [B] and [EO] that this singularity has a genus–one open book decomposition supporting that contact structure
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