## Algebraic & Geometric Topology

### Planar open books with four binding components

Yankı Lekili

#### Abstract

We study an explicit construction of planar open books with four binding components on any three-manifold which is given by integral surgery on three component pure braid closures. This construction is general, indeed any planar open book with four binding components is given this way. Using this construction and results on exceptional surgeries on hyperbolic links, we show that any contact structure of $S3$ supports a planar open book with four binding components, determining the minimal number of binding components needed for planar open books supporting these contact structures. In addition, we study a class of monodromies of a planar open book with four binding components in detail. We characterize all the symplectically fillable contact structures in this class and we determine when the Ozsváth–Szabó contact invariant vanishes. As an application, we give an example of a right-veering diffeomorphism on the four-holed sphere which is not destabilizable and yet supports an overtwisted contact structure. This provides a counterexample to a conjecture of Honda, Kazez and Matić from [J. Differential Geom. 83 (2009) 289–311].

#### Article information

Source
Algebr. Geom. Topol., Volume 11, Number 2 (2011), 909-928.

Dates
Accepted: 9 January 2011
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715213

Digital Object Identifier
doi:10.2140/agt.2011.11.909

Mathematical Reviews number (MathSciNet)
MR2782547

Zentralblatt MATH identifier
1220.57017

Subjects
Primary: 57R17: Symplectic and contact topology

#### Citation

Lekili, Yankı. Planar open books with four binding components. Algebr. Geom. Topol. 11 (2011), no. 2, 909--928. doi:10.2140/agt.2011.11.909. https://projecteuclid.org/euclid.agt/1513715213

#### References

• C Adams, W Sherman, Minimum ideal triangulations of hyperbolic $3$–manifolds, Discrete Comput. Geom. 6 (1991) 135–153
• L Armas-Sanabria, M Eudave-Muñoz, The hexatangle, Topology Appl. 156 (2009) 1037–1053
• J Baldwin, Capping off open books and the Ozsváth–Szabó contact invariant
• J Baldwin, J B Etnyre, A note on the support norm of a contact structure, to appear in “Proceedings of the 2009 Georgia Topology Conference (Athens, GA)”
• T Etgü, Y Lekili, Examples of planar tight contact structures with support norm one, Int. Math. Res. Not. (2010) 3723–3728
• J B Etnyre, Planar open book decompositions and contact structures, Int. Math. Res. Not. (2004) 4255–4267
• J B Etnyre, Lectures on open book decompositions and contact structures, from: “Floer homology, gauge theory, and low-dimensional topology”, (D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó, editors), Clay Math. Proc. 5, Amer. Math. Soc. (2006) 103–141
• J B Etnyre, B Ozbagci, Invariants of contact structures from open books, Trans. Amer. Math. Soc. 360 (2008) 3133–3151
• E Fadell, L Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111–118
• R Fox, L Neuwirth, The braid groups, Math. Scand. 10 (1962) 119–126
• P Ghiggini, On tight contact structures with negative maximal twisting number on small Seifert manifolds, Algebr. Geom. Topol. 8 (2008) 381–396
• E Giroux, Géométrie de contact: de la dimension trois vers les dimensions supérieures, from: “Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002)”, (T Li, editor), Higher Ed. Press, Beijing (2002) 405–414
• C M Gordon, Small surfaces and Dehn filling, from: “Proceedings of the Kirbyfest (Berkeley, CA, 1998)”, (J Hass, M Scharlemann, editors), Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 177–199
• C M Gordon, Y-Q Wu, Toroidal and annular Dehn fillings, Proc. London Math. Soc. $(3)$ 78 (1999) 662–700
• K Honda, W H Kazez, G Matić, Right-veering diffeomorphisms of compact surfaces with boundary, Invent. Math. 169 (2007) 427–449
• K Honda, W H Kazez, G Matić, On the contact class in Heegaard Floer homology, J. Differential Geom. 83 (2009) 289–311
• W Magnus, A Peluso, On knot groups, Comm. Pure Appl. Math. 20 (1967) 749–770
• B Martelli, C Petronio, Dehn filling of the “magic” $3$–manifold, Comm. Anal. Geom. 14 (2006) 969–1026
• K Niederkrüger, C Wendl, Weak symplectic fillings and holomorphic curves, to appear in Ann. Sci. École Norm. Sup.
• P Ozsváth, A Stipsicz, Z Szabó, Planar open books and Floer homology, Int. Math. Res. Not. (2005) 3385–3401
• P Ozsváth, Z Szabó, Heegaard Floer homology and contact structures, Duke Math. J. 129 (2005) 39–61
• V V Prasolov, A B Sossinsky, Knots, links, braids and $3$–manifolds. An introduction to the new invariants in low-dimensional topology, Translations of Math. Monogr. 154, Amer. Math. Soc. (1997) Translated from the Russian manuscript by Sossinsky
• W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
• A Wand, Mapping class group relations, Stein fillings, and planar open book decompositions
• C Wendl, Strongly fillable contact manifolds and $J$–holomorphic foliations, Duke Math. J. 151 (2010) 337–384