• with S. Sivek, Fillings of unit cotangent bundles, 14pp., arXiv:1510.06736,

submitted.

http://arxiv.org/abs/1510.06736

Abstract: We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $\Sigma_g$, where g is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $\Sigma_g$. For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values: for example, if $g−1$ is square-free, then any exact filling has the same integral homology and intersection form as $DT^*\Sigma_g$.

  • with C. Kutluhan, G. Matic, and A. Wand, Algebraic torsion via Heegaard Floer

homology, 10pp., arXiv:1503.01685.

http://arxiv.org/abs/1503.01685

Abstract: We use Hutchings's prescription to define the notion of algebraic torsion for closed contact 3-manifolds via Heegaard Floer homology. Then we show that a closed 3-manifold equipped with an overtwisted contact structure has algebraic torsion of order zero.

  • with R. I. Baykur and N. Monden, Positive factorizations of mapping classes,

22pp., arXiv:1412.0352.

http://arxiv.org/abs/1412.0352

Abstract: In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whether or not there exist genus-g Lefschetz pencils with n base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. We observe that only symplectic 4-manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.

  • with P. Ghiggini and K. Honda, The vanishing of the contact invariant in the

presence of torsion, 6pp., arXiv:0706.1602.

http://arxiv.org/abs/0706.1602

Abstract: We prove that the Ozsvath-Szabo contact invariant of a closed contact 3-manifold with positive Giroux torsion vanishes.

  • with J. B. Etnyre, Monoids in the mapping class group, 36pp., arXiv:1504.02106. To appear in Interactions between low dimensional topology and mapping class groups, Bonn Conference Proceedings, Geometry and Topology Monographs.

http://arxiv.org/abs/1504.02106

Abstract:  In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.

  • with R. I. Baykur, Topological complexity of symplectic 4-manifolds and Stein fillings, 25pp., arXiv:1212.1699, to appear in J. Symplectic Geom.

http://arxiv.org/abs/1212.1699

Abstract: We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book, except possibly for a few low-genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products

of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by Auroux, Smith and Wajnryb, and a more general variant of it raised by Korkmaz, Ozbagci and Stipsicz, independently.

  • with R. I. Baykur, S. Lisi, and C. Wendl, Families of contact 3-manifolds with

arbitrarily large Stein fillings, J. Differential Geom. Volume 101, Number 3 (2015), 423-465.

http://projecteuclid.org/euclid.jdg/1445518920

Abstract: We show that there are vast families of contact 3-manifolds, each member of which admits infinitely many Stein fillings with arbitrarily large Euler characteristics and arbitrarily small signatures—disproving a conjecture of Stipsicz and Ozbagci. To produce our examples, we use a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in [24].

  • with P. Ghiggini, Tight contact structures on the Brieskorn spheres \Sigma(2, 3, 6n-1) and contact invariants, to appear in J. Reine Angew. Math. (Crelle's Journal) ISSN (Online) 1435-5345, DOI: 10.1515/crelle-2014-0038, June 2014.

http://www.degruyter.com/view/j/crelle.ahead-of-print/crelle-2014-0038/crelle-2014-0038.xml

Abstract: We compute the Ozsváth–Szabó contact invariants for all tight contact structures on the manifolds -\Sigma(2,3,6n-1) using twisted coefficients and a previous computation by the first author and Ko Honda. This computation completes the classification of the tight contact structures in this family of 3-manifolds.

  • with K. L. Baker and J. B. Etnyre, Cabling, contact structures and mapping class monoids, J. Differential Geom. 90 (2012), no. 1, 1-80;

http://projecteuclid.org/euclid.jdg/1335209489

Abstract: In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

  • with H. Endo and T. Mark, Monodromy Substitutions and Rational Blowdowns, J. Topology, 4 (2012), no.1, 227-253;

http://jtopol.oxfordjournals.org/content/4/1/227.abstract

Abstract: We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of rational blowdowns of 4-manifolds, specifically via monodromy substitution in Lefschetz fibrations. The simplest example is the lantern relation, already shown by the first author and Gurtas (`Lantern relations and rational blowdowns', Proc. Amer. Math. Soc. 138 (2010) 1131--1142) to correspond to rational blowdown along a (-4)-sphere; here we give relations that extend that result to realize the `generalized' rational blowdowns of Fintushel and Stern (`Rational blowdowns of smooth 4-manifolds,' J. Differential Geom. 46 (1997) 181--235) and Park (`Seiberg-Witten invariants of generalised rational blow-downs,' Bull. Austral. Math. Soc. 56 (1997) 363--384) by monodromy substitution, as well as several of the families of rational blowdowns discovered by Stipsicz, Szabo, and Wahl (`Rational blowdowns and smoothings of surface singularities,' J. Topol. 1 (2008) 477--517).

  • with J. B. Etnyre, Fibered Transverse Knots and the Bennequin Bound, Int. Math. Res. Not. (2011) no. 7, 1483-1509;

http://imrn.oxfordjournals.org/content/2011/7/1483.abstract

Abstract: We prove that a nicely fibered link (by which we mean the binding of an open book) in a tight contact manifold (M, \xi) with zero Giroux torsion has a transverse representative realizing the Bennequin bound if and only if the contact structure it supports (since it is also the binding of an open book) is \xi. This gives a geometric reason for the non-sharpness of the Bennequin bound for fibered links. We also give the classification, up to contactomorphism, of maximal self-linking

number links in these knot types. Moreover, in the standard tight contact structure on S^3 we classify, up to transverse isotopy, transverse knots with maximal self-linking number in the knot types given as closures of positive braids and given as fibered strongly quasi-positive knots. We derive several braid theoretic corollaries from this. In particular, we give conditions under which quasi-positive braids are related by positive Markov stabilizations and when a minimal braid index representative of a knot is quasi-positive. Finally, we observe that our main result can be used to

show, and make rigorous the statement, that contact structures on a given manifold are in a strong sense classified by the transverse knot theory they support.

  • with O. Plamenevskaya, Planar open books, monodromy factorizations, and symplectic fillings, Geom. Topol. 14 (2010) no.4, 2077-2101;

http://msp.org/gt/2010/14-4/p06.xhtml

Abstract: We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on $L(p,1)$ ($p \neq 4$) has a unique filling, and describe fillable and non-fillable tight contact structures on certain Seifert fibered spaces.