Milnor open books of links of some rational surface singularities

TL;DR

This paper explores the structure of Milnor open books and Legendrian surgery diagrams for certain rational surface singularities, revealing differences between Milnor and support invariants in an infinite family of contact 3-manifolds.

Contribution

It provides explicit descriptions of Milnor open books and Legendrian surgery diagrams for specific rational surface singularities and introduces a family where Milnor and support invariants differ.

Findings

Milnor open books and Legendrian diagrams are characterized for these singularities.

An infinite family of contact 3-manifolds exhibits Milnor genus greater than support genus.

The paper demonstrates the strict inequality between Milnor and support invariants in this family.

Abstract

We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. support norm) of the canonical contact structure, for each member of this family.