Volume 17, issue 2 (2013)

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Knot contact homology

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

Geometry & Topology 17 (2013) 975–1112
Abstract

The conormal lift of a link $K$ in ${ℝ}^{3}$ is a Legendrian submanifold ${\Lambda }_{K}$ in the unit cotangent bundle ${U}^{\ast }{ℝ}^{3}$ of ${ℝ}^{3}$ 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 ${\Lambda }_{K}$, the homology of a differential graded algebra generated by Reeb chords whose differential counts holomorphic disks in the symplectization $ℝ×{U}^{\ast }{ℝ}^{3}$ with Lagrangian boundary condition $ℝ×{\Lambda }_{K}$.

We perform an explicit and complete computation of the Legendrian homology of ${\Lambda }_{K}$ 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.

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