## Kyoto Journal of Mathematics

### Contact structures on plumbed 3-manifolds

Çağrı Karakurt

#### Abstract

We show that the Ozsváth–Szabó contact invariant $c^{+}(\xi)\inHF^{+}(-Y)$ of a contact $3$-manifold $(Y,\xi)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$ and if $\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó 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 Ozsváth–Stipsicz–Szabó obstruction to admitting a planar open book for a certain class of contact structures. We also define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. Using this technique, we do a sample calculation showing that the invariant can get infinitely many distinct values.

#### Article information

Source
Kyoto J. Math., Volume 54, Number 2 (2014), 271-294.

Dates
First available in Project Euclid: 2 June 2014

https://projecteuclid.org/euclid.kjm/1401741279

Digital Object Identifier
doi:10.1215/21562261-2642395

Mathematical Reviews number (MathSciNet)
MR3215568

Zentralblatt MATH identifier
1300.57026

#### Citation

Karakurt, Çağrı. Contact structures on plumbed 3-manifolds. Kyoto J. Math. 54 (2014), no. 2, 271--294. doi:10.1215/21562261-2642395. https://projecteuclid.org/euclid.kjm/1401741279

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