Milnor open books of links of some rational surface singularities

Abstract

We determine Legendrian surgery diagrams for the canonical contact structures of links of rational surface singularities that are also small Seifert fibered 3-manifolds. Moreover, we describe an infinite family of Milnor fillable contact 3-manifolds so that, for each member of this family, the Milnor genus and Milnor norm are strictly greater than the support genus and support norm of the canonical contact structure. For some of these contact structures we construct supporting Milnor open books.

Complex singularities and contact topology

“…An affine manifold of dimension 1 A complex manifold of dimension 1 (a real affine curve) (a complex curve) (11) A real affine manifold A complex manifold (12) An affine map from U ⊂ R A holomorphic map from U ⊂ C to a real affine manifold to a complex manifold (13) An affine real-valued function A holomorphic complex-valued function on a real affine manifold on a complex manifold (14) An affine real-valued function A pluriharmonic real-valued function on a real affine manifold on a complex manifold (15) The condition Hess ρ ≥ 0 The condition −dd c ρ ≥ 0 on a real affine manifold on a complex manifold (16) A (strictly) convex function A (strictly) plurisubharmonic function on a real affine manifold on a complex manifold As the total space of the tangent bundle TRP 2 retracts by deformation onto RP 2 , they have the same second Betti number. As RP 2 is a non-orientable surface, we see that H 2 (RP 2 , Z) = 0.…”

“…• The conditions of line (15) are generalizations to manifolds of arbitrary dimension of the conditions of line (8). In fact, in both cases a real-valued function ρ satisfies them identically if and only if it satisfies the corresponding condition (8) in restriction to any geodesic (map of the kind described in line (12)).…”

Contact structures and reducible surgeries

“…There is a canonical contact structure ξ can on L(p, q), defined as follows: the standard contact structure ξ std on S 3 is Z/pZ-equivariant under the action used to define L(p, q), and ξ can is defined as the quotient of ξ std under this action. We know that ξ can is the contact structure ξ |a 1 |−2,|a 2 |−2,...,|an|−2 , in which each r i is as large as possible [58, Proposition 3.2] (see also [8,Section 7]).…”

“…The inequality p ≤ τ + 2 implies that p ≤ 2t − 4 for all τ ≥ 7, so this leaves only τ ≤ 6, in which case p ≤ τ + 2 ≤ 8 and the only such lens space is L (8,3). In this case we have d 3 (ξ can ) = − 1 4 and so the induced contact structure must be ξ can or its conjugate, by Proposition 4.2, with a Stein filling W having intersection form −8 by Proposition 1.4.…”

“…In this case we have d 3 (ξ can ) = − 1 4 and so the induced contact structure must be ξ can or its conjugate, by Proposition 4.2, with a Stein filling W having intersection form −8 by Proposition 1.4. Lisca [47] showed that W must be diffeomorphic to a blow-up of one of two fillings, denoted W 8,3 (1, 2, 1) or W 8,3 (2, 1, 2) (note that 8 8−3 = [2, 3, 2]), and the first cannot occur since it has b 2 = 2. The second is constructed from a diagram in which 2-handles are attached to a pair of parallel −1-framed unknots; the cocores of these handles, together with the annulus they cobound, produce a sphere of self-intersection −2, and so the intersection form of this or any blow-up cannot be −8 .…”