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We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration., Comment: Changes made following referee's suggestions. One reference and a few extra clarifying comments also added. To appear in Journal of Symplectic Geometry

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Nemethi. We also show that the construction cannot always work in the reverse direction: in fact the U-filtration depth of contact Ozsvath-Szabo invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsvath-Szabo invariants of those contact structures fillable by a AR plumbing graph, generalizing an earlier work of the first author., Comment: 17 pages, 8 figures

Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks., Comment: 10 pages, 8 figures

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times., Comment: 35 pages, 20 figures; v2 has significantly more examples of small exotic symplectic 4-manifolds including exotic CP2 blown up 6 and 7 times and the structure of the paper is reorganized; v3 corrects minor errors and some inconsistencies in conventions

We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsvath--Szabo invariants can be calculated algorithmically. We run this algorithm for the first few members of the family and list the resulting Heegaard--Floer homologies. We also show that the rank of the Heegaard--Floer homology can get arbitrary large values in this family by using its relation with the Casson invariant. For comparison, we list the ranks of Floer homologies of all the examples of Briekorn spheres that are known to bound contractible manifolds., Comment: 7 Figures, 2 Tables

In this paper, we show that the Ozsv\'ath-Szab\'o contact invariant $c^+(\xi)\in HF^+(-Y)$ of a contact 3-manifold $(Y,\xi)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsv\'ath and Szab\'o to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsv\'ath-Stipsicz-Szab\'o obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values., Comment: Added some examples and comments