## Geometry & Topology

### Monopole Floer homology and Legendrian knots

Steven Sivek

#### Abstract

We use monopole Floer homology for sutured manifolds to construct invariants of unoriented Legendrian knots in a contact $3$–manifold. These invariants assign to a knot $K⊂Y$ elements of the monopole knot homology $KHM(−Y,K)$, and they strongly resemble the knot Floer homology invariants of Lisca, Ozsváth, Stipsicz, and Szabó. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of nonloose knots in overtwisted $3$–manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

#### Article information

Source
Geom. Topol., Volume 16, Number 2 (2012), 751-779.

Dates
Revised: 3 February 2012
Accepted: 30 January 2012
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732408

Digital Object Identifier
doi:10.2140/gt.2012.16.751

Mathematical Reviews number (MathSciNet)
MR2928982

Zentralblatt MATH identifier
1270.57050

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology
Secondary: 57R17: Symplectic and contact topology

#### Citation

Sivek, Steven. Monopole Floer homology and Legendrian knots. Geom. Topol. 16 (2012), no. 2, 751--779. doi:10.2140/gt.2012.16.751. https://projecteuclid.org/euclid.gt/1513732408

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