Knot contact homology
Abstract
The conormal lift of a link K in ℝ3is a Legendrian submanifold ∧Kin the unit cotangent
bundle U*ℝ3of ℝ3with 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 ∧K, the homology of a differential graded algebra generated by Reeb chords
whose differential counts holomorphic disks in the symplectization R × U*ℝ3with Lagrangian
boundary condition R × ∧K. We perform an explicit and complete computation of the
Legendrian homology of ∧Kfor 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.
Type
Journal articleSubject
Science & TechnologyPhysical Sciences
Mathematics
LEGENDRIAN SUBMANIFOLDS
BRAID INVARIANTS
SURGERY
IMMERSIONS
EMBEDDINGS
R2N+1
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https://hdl.handle.net/10161/17788Published Version (Please cite this version)
10.2140/gt.2013.17.975Publication Info
Ekholm, T; Etnyre, JB; Ng, L; & Sullivan, MG (2013). Knot contact homology. Geometry & Topology, 17(2). pp. 975-1112. 10.2140/gt.2013.17.975. Retrieved from https://hdl.handle.net/10161/17788.This is constructed from limited available data and may be imprecise. To cite this
article, please review & use the official citation provided by the journal.
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Show full item recordScholars@Duke
Lenhard Lee Ng
Eads Family Professor
My research mainly focuses on symplectic topology and low-dimensional topology. I
am interested in studying structures in symplectic and contact geometry (Weinstein
manifolds, contact manifolds, Legendrian and transverse knots), especially through
holomorphic-curve techniques. One particular interest is extracting topological information
about knots through cotangent bundles, and exploring relations to topological string
theory. I have also worked in Heegaard Floer theory, quantum topology, and

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