Holomorphic disks and knot invariants

Details Holomorphic disks and knot invariants Holomorphic disks and knot invariants

2002-09-06

arxiv.org/abs/math/0209056v4

Adv. Math. 186 (2004), no. 1, 58--116.

Mathematics

Geometric Topology

64 pages, 17. Revised version

Scientific paper

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 $HF^+$ for surgeries along the knot. Applications include calculation of $HF^+$ of three-manifolds obtained by surgeries on some special knots in $S^3$, and also calculation of $HF^+$ for certain simple three-manifolds which fiber over the circle.

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