Surgery diagrams for horizontal contact structures

Abstract

ABSTRACT. We describe Legendrian surgery diagrams for some horizontal contact structures on non-positive plumbing trees of oriented circle bundles over spheres with negative Euler numbers. As an application we determine Milnor fillable contact structures on some Milnor fillable 3-manifolds.

Symplectic fillings of lens spaces as Lefschetz fibrations

“…By the proof of Theorem 4, the induced open book on the boundary is fixed for all distinct PALFs constructed for a given lens space. The desired result follows since we know that the induced open book on the boundary of the canonical PALF on the minimal resolution is compatible with the canonical contact structure [14].…”

Milnor open books of links of some rational surface singularities

“…Note that in the star-shaped plumbing tree of Y all the weights are less than or equal to −2 and therefore one can Legendrian realize these unknots to obtain distinct Stein fillable contact structures on Y . The construction in [32], coupled with Wu's classification [38], allows us to conclude that Proposition 12. The canonical contact structure ξ can on Y = Y (e 0 ; r 1 , r 2 , r 3 ), where e 0 ≤ −3, can be identified as the contact structure obtained by putting all the extra zigzags of the Legendrian unknots in the star-shaped presentation of Y , to one fixed side.…”

“…, n. The next result immediately follows from Proposition 9. (Note that a Legendrian surgery diagram for ξ can on L(p, q) is given by Figure 4 in [32].) Corollary 11.…”

“…Remark 10. The canonical contact structure ξ can of the link of a singularity as in Proposition 9 can be explicitly given by a Legendrian surgery diagram using the methods discussed in [32].…”

“…It is well-known [31] that the small Seifert fibred 3-manifold Y = Y (e 0 ; r 1 , r 2 , r 3 ) can also be described by a star-shaped weighted plumbing tree, where the central vertex has weight e 0 and there are three legs emanating from that vertex corresponding to the continued fraction expansions of the rational numbers − 1 r i , for i = 1, 2, 3, respectively. If e 0 ≤ −3, then we can easily identify ξ can on Y , based on the work in [13,32,38] (see Section 6).…”