Classification of tight contact structures on surgeries on the figure-eight knot
- with Hyunki Min
- PDF — arXiv — Journal: Geometry & Topology Vol. 24 (2020), Issue 3, 1457–1517
- Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
Contact surgery and symplectic caps
- with John B. Etnyre
- PDF — arXiv — Journal: Bulletin of the LMS, Vol. 52 (2020), Issue 2, 379–394
- In this note we show that a closed oriented contact manifold is obtained from the standard contact sphere of the same dimension by contact surgeries on isotropic and coisotropic spheres. In addition, we observe that all closed oriented contact manifolds admit symplectic caps.
Mazur-type manifolds with \(L\)-space boundary
- with Bülent Tosun
- PDF — arXiv — Journal: MRL, Vol. 27 (2020), No. 1, 35–42
- In this note, we prove that if the boundary of a Mazur-type \(4\)-manifold is an irreducible Heegaard Floer homology \(L\)-space, then the manifold must be the \(4\)-ball, and the boundary must be the \(3\)-sphere. We use this to give a new proof of Gabai's Property R.
Symplectic fillings, contact surgeries, and Lagrangian disks
- with John B. Etnyre and Bülent Tosun
- PDF — arXiv — Journal: IMRN, Vol. 2021, Issue 8, April 2021, 6020–6050
- This paper completely answers the question of when contact \((r)\)-surgery on a Legendrian knot in the standard contact structure on the 3-sphere yields a symplectically fillable contact manifold for \(r\) in \((0,1]\). We also give obstructions for other positive \(r\) and investigate Lagrangian fillings of Legendrian knots.
Tight Contact Structures via Admissible Transverse Surgery
- PDF — arXiv — Journal: Journal of Knot Theory and Its Ramifications, Vol. 28, No. 04, 1950032 (2019)
- We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure \(\xi_K\) is supported by the fibred knot \(K\subset M\), we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure \(\xi_\overline{K}\) supported by the mirror knot \(\overline{K} \subset M\). We derive several corollaries about the existence of tight contact structures, L-space knots outside \(S^3\), non-planar contact structures, and non-planar Legendrian knots.
Contact Surgeries on the Legendrian Figure-Eight Knot
Overtwisted Positive Contact Surgeries
Transverse Surgery on Knots in Contact 3-Manifolds
- PDF — arXiv (Sections 1-5) — Journal: Transactions of the AMS, 372 (2019), 1671–1707
- We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. One main aim of this paper is to show that in many contexts, transverse surgery is a more natural tool than surgery on Legendrian knots. We reinterpret contact \((\pm 1)\)-surgery on Legendrian knots as transverse surgery on transverse push-offs, allowing us to give simpler proofs of known results. We give the first result on the tightness of inadmissible transverse surgery (cf. contact \((+1)\)-surgery) for contact manifolds with vanishing Heegaard Floer contact invariant. In particular, inadmissible transverse \(r\)-surgery on the connected binding of a genus \(g\) open book that supports a tight contact structure preserves tightness if \(r > 2g-1\).
Tight Planar Contact Manifolds with Vanishing Heegaard Floer Contact Invariants
- with Amey Kaloti and Dheeraj Kulkarni
- PDF — arXiv — Journal: Topology and its Applications 212 (2016), 19–28
- In this note, we exhibit infinite families of tight non-fillable contact manifolds supported by planar open books with vanishing Heegaard Floer contact invariants. Moreover, we also exhibit an infinite such family where the supported manifold is hyperbolic.
