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2013 Knot contact homology
Tobias Ekholm, John B Etnyre, Lenhard Ng, Michael G Sullivan
Geom. Topol. 17(2): 975-1112 (2013). DOI: 10.2140/gt.2013.17.975

Abstract

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.

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Tobias Ekholm. John B Etnyre. Lenhard Ng. Michael G Sullivan. "Knot contact homology." Geom. Topol. 17 (2) 975 - 1112, 2013. https://doi.org/10.2140/gt.2013.17.975

Information

Received: 16 January 2012; Accepted: 5 January 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1267.53095
MathSciNet: MR3070519
Digital Object Identifier: 10.2140/gt.2013.17.975

Subjects:
Primary: 53D42
Secondary: 57M27 , 57R17

Keywords: contact homology , holomorphic curves , knot invariants , Legendrian submanifolds

Rights: Copyright © 2013 Mathematical Sciences Publishers

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Vol.17 • No. 2 • 2013
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