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Knot contact homology

Tobias Ekholm, John B Etnyre, Lenhard Ng and Michael G Sullivan

Geometry & Topology 17 (2013) 975–1112

The conormal lift of a link K in 3 is a Legendrian submanifold ΛK in the unit cotangent bundle U3 of 3 with contact structure equal to the kernel of the Liouville form. Knot contact homology, a topological link invariant of K, is defined as the Legendrian homology of ΛK, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization × U3 with Lagrangian boundary condition × ΛK.

We perform an explicit and complete computation of the Legendrian homology of ΛK for arbitrary links K in terms of a braid presentation of K, confirming a conjecture that this invariant agrees with a previously defined combinatorial version of knot contact homology. The computation uses a double degeneration: the braid degenerates toward a multiple cover of the unknot, which in turn degenerates to a point. Under the first degeneration, holomorphic disks converge to gradient flow trees with quantum corrections. The combined degenerations give rise to a new generalization of flow trees called multiscale flow trees. The theory of multiscale flow trees is the key tool in our computation and is already proving to be useful for other computations as well.

contact homology, holomorphic curves, knot invariants, Legendrian submanifolds
Mathematical Subject Classification 2010
Primary: 53D42
Secondary: 57R17, 57M27
Received: 16 January 2012
Accepted: 5 January 2013
Published: 30 April 2013
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Ralph Cohen
Tobias Ekholm
Department of Mathematics
Uppsala Unversity
Box 480
751 06 Uppsala
John B Etnyre
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta GA 30332-0160
Lenhard Ng
Department of Mathematics
Duke University
Box 90320
Durham NC 27708-0320
Michael G Sullivan
Department of Mathematics and Statistics
University of Massachusetts
Amherst MA 01003-9305