Holomorphic disks and knot invariants

Abstract

We define a Floer-homology invariant for knots in an oriented three-manifold, closely related to the holomorphic disk Floer homologies for three-manifolds defined in an earlier paper. We set up basic properties of these invariants, including an Euler characteristic calculation, behaviour under connected sums. Then, we establish a relationship with $H{F}^{+}$ for surgeries along the knot. Applications include calculation of $H{F}^{+}$ of three-manifolds obtained by surgeries on some special knots in $S^3$, and also calculation of $H{F}^{+}$ for certain simple three-manifolds which fiber over the circle.

64 pages, 17. Revised version

References in corpus (6)

• Floer homology and knot complements
• Heegaard Floer homology and alternating knots
• Holomorphic disks and three-manifold invariants: properties and applications
• Heegaard Floer homologies and contact structures
• Holomorphic triangles and invariants for smooth four-manifolds
• Knot Floer homology, genus bounds, and mutation

Cited by in corpus (7)

• Floer homology and knot complements
• Heegaard Floer homologies and contact structures
• Khovanov homology and the slice genus
• Monopoles and lens space surgeries
• Knot Floer homology, genus bounds, and mutation
• Knots with unknotting number one and Heegaard Floer homology
• L-space knots have no essential Conway spheres