# Legendrian knots and monopoles

@article{Mrowka2004LegendrianKA, title={Legendrian knots and monopoles}, author={Tomasz S. Mrowka and Yann Rollin}, journal={Algebraic \& Geometric Topology}, year={2004}, volume={6}, pages={1-69} }

We prove a generalization of Bennequin’s inequality for Legendrian knots in a 3‐ dimensional contact manifold.Y;/ , under the assumption that Y is the boundary of a 4‐dimensional manifold M and the version of Seiberg‐Witten invariants introduced by Kronheimer and Mrowka in [10] is nonvanishing. The proof requires an excision result for Seiberg‐Witten moduli spaces; then the Bennequin inequality becomes a special case of the adjunction inequality for surfaces lying inside M . 57R17, 57M25, 57M27…

## 38 Citations

### Seiberg-Witten Floer homotopy contact invariant

- Mathematics
- 2020

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer-Mrowka-Ozv\'ath-Szab\'o. Moreover, we prove a gluing formula relating our invariant with the first author's…

### Monopoles and foliations without holonomy-invariant transverse measure

- MathematicsJournal of Symplectic Geometry
- 2022

This article proves a uniform exponential decay estimate for SeibergWitten equations on non-compact 4-manifolds with exact symplectic ends of bounded geometry. This is an extension of the analysis…

### A Bauer–Furuta-type refinement of Kronheimer and Mrowka’s invariant for 4–manifolds with contact boundary

- MathematicsAlgebraic & Geometric Topology
- 2021

Kronheimer and Mrowka constructed a variant of Seiberg-Witten invariants for a 4-manifold $X$ with contact boundary in 1997. Using Furuta's finite dimensional approximation, we refine this invariant…

### Involutions, knots, and Floer K-theory

- Mathematics
- 2021

We establish a version of Seiberg–Witten FloerK-theory for knots, as well as a version of Seiberg–Witten Floer K-theory for 3-manifolds with involutions. The main theorem is a 10/8-type inequality…

### An adjunction criterion in almost-complex 4-manifolds

- Mathematics
- 2021

The adjunction inequality is a key tool for bounding the genus of smoothly embedded surfaces in 4-manifolds. Using gauge-theoretic invariants, many versions of this inequality have been established…

### A note on generalized Thurston--Bennequin inequalities

- Mathematics
- 2022

. We give a generalized Thurston–Bennequin-type inequality for links in S 3 using a Bauer–Furuta-type invariant for 4-manifolds with contact boundary. As a special case, we also give an adjunction…

### Instanton Floer homology and contact structures

- MathematicsSelecta Mathematica
- 2015

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or…

### Instanton Floer homology and contact structures

- Mathematics
- 2016

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka’s sutured instanton Floer homology theory. This is the first invariant of contact manifolds—with or…

### On the Slicing Genus of Legendrian Knots

- Mathematics
- 2005

We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.

### Exotic Dehn twists on sums of two contact 3-manifolds

- Mathematics
- 2022

We exhibit the first examples of exotic contactomorphisms with infinite order as elements of the contact mapping class group. These are given by certain Dehn twists on the separating sphere in a…

## References

SHOWING 1-10 OF 20 REFERENCES

### THE SEIBERG-WITTEN INVARIANTS AND SYMPLECTIC FORMS

- Mathematics
- 1994

(Note: There are no symplectic forms on X unless b and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for the…

### An infinite family of tight, not semi-fillable contact three-manifolds

- Mathematics
- 2003

We prove that an innite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi{llings. The argument uses results…

### Handlebody construction of Stein surfaces

- Mathematics
- 1998

The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained-they correspond to…

### The Genus of Embedded Surfaces in the Projective Plane

- Mathematics
- 1994

1. Statement of the result The genus of a smooth algebraic curve of degree d in CP is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the…

### TOPOLOGICAL CHARACTERIZATION OF STEIN MANIFOLDS OF DIMENSION >2

- Mathematics
- 1990

In this paper I give a completed topological characterization of Stein manifolds of complex dimension >2. Another paper (see [E14]) is devoted to new topogical obstructions for the existence of a…

### Gluing tight contact structures

- Mathematics
- 2001

We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a…

### Geometry of Low-dimensional Manifolds: Filling by holomorphic discs and its applications

- Mathematics
- 1991

The survey is devoted to application of the technique of filling by holomorphic discs to different symplectic and complex analytic problems. COMPLEX AND SYMPLECTIC RECOLLECTIONS J -Convexity Let X, J…

### Few remarks about symplectic filling

- Mathematics
- 2003

We show that any compact symplectic manifold $W,\omega$ with boundary embeds as a domain into a closed symplectic manifold, provided that there exists a contact plane $\xi$ on $\partial W$ which is…

### Tight, not semi–fillable contact circle bundles

- Mathematics
- 2002

Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi–fillable, thus confirming a conjecture of Ko Honda.