Knot contact homology

dc.contributor.author

Ekholm, T

dc.contributor.author

Etnyre, JB

dc.contributor.author

Ng, L

dc.contributor.author

Sullivan, MG

dc.date.accessioned

2018-12-11T15:21:32Z

dc.date.available

2018-12-11T15:21:32Z

dc.date.issued

2013-04-30

dc.date.updated

2018-12-11T15:21:31Z

dc.description.abstract

The conormal lift of a link K in ℝ3is a Legendrian submanifold ∧Kin the unit cotangent bundle Uℝ3of 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 × Uℝ3with 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.

dc.identifier.issn

1465-3060

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1364-0380

dc.identifier.uri

https://hdl.handle.net/10161/17788

dc.language

English

dc.publisher

Mathematical Sciences Publishers

dc.relation.ispartof

Geometry & Topology

dc.relation.isversionof

10.2140/gt.2013.17.975

dc.subject

Science & Technology

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Physical Sciences

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Mathematics

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LEGENDRIAN SUBMANIFOLDS

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BRAID INVARIANTS

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SURGERY

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IMMERSIONS

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EMBEDDINGS

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R2N+1

dc.title

Knot contact homology

dc.type

Journal article

pubs.begin-page

975

pubs.end-page

1112

pubs.issue

2

pubs.organisational-group

Trinity College of Arts & Sciences

pubs.organisational-group

Duke

pubs.organisational-group

Mathematics

pubs.publication-status

Published

pubs.volume

17

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