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Register a new userSome applications of Menkes JSJ decomposition for symplectic fillings
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We apply Menkes JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a symplectic 2-handle to a filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying symplectic fillings to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.
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We prove that if a contact manifold $(M,xi)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,xi)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces.
We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.
In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.
In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,xi_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,xi_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.
In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $pi$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $pi$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually overtwisted tight contact structure on the negative parabolic torus bundle with monodromy $-T^n$ ($n<0$), we prove that it is Stein fillable. By the way, we give a partial answer to a conjecture of Golla and Lisca.
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