Knot contact homology

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

doi:10.2140/gt.2013.17.975

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Knot contact homology

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

Geometry & Topology 17 (2013) 975–1112

DOI: 10.2140/gt.2013.17.975

Abstract

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

We perform an explicit and complete computation of the Legendrian homology of $Λ_{K}$ 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

References

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/

Volume 17, issue 2 (2013)