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2014 | OriginalPaper | Buchkapitel

A Topological Introduction to Knot Contact Homology

verfasst von : Lenhard Ng

Erschienen in: Contact and Symplectic Topology

Verlag: Springer International Publishing

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Abstract

Knot contact homology is a Floer-theoretic knot invariant derived from counting holomorphic curves in the cotangent bundle of \(\mathbb{R}^{3}\) with Lagrangian boundary condition on the conormal bundle to the knot. Among other things, this can be used to produce a three-variable polynomial that detects the unknot and conjecturally contains many known knot invariants; a different part of the package yields an effective invariant of transverse knots in \(\mathbb{R}^{3}\).

In these notes we will describe knot contact homology and the geometry and algebra behind it, as well as connections to other knot invariants, transverse knot theory, and physics. Topics to be treated along the way include: Legendrian contact homology for Legendrian knots in R ³; the conormal construction and Legendrian contact homology in five dimensions; a combinatorial formulation of knot contact homology in terms of braids; transverse homology, a filtered version associated to transverse knots; and relations to the HOMFLY polynomial (and knot homologies) and to recent work of Vafa and others in string theory.

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Vorheriges Kapitel Lecture Notes on Embedded Contact Homology

Nur mit Berechtigung zugänglich

To define this homology class, we assume that “capping half-disks” have been chosen in V for each Reeb chord a _i, with boundary given by a _i along with a path in Λ joining the endpoints of a _i. Some additional care must be taken if Λ has multiple components.

The profusion of terms and specializations is an unfortunate byproduct of the way that the subject evolved over a decade.

See the Appendix for differences in convention between our definition and the ones from [34] and [15].

I.e., no repeated factors.

Caution: the polynomial described here differs from the augmentation polynomial from [34] by a change of variables μ↦−1/μ. See the Appendix.

In a related vein, Fuji, Gukov, and Sulkowski [20] have proposed a four-variable “super-A-polynomial” that specializes to the Q-deformed A-polynomial.

The transverse representatives of m(7₆), 9₄₄, 9₄₈, 10₁₃₆, and 10₁₆₀ cannot be distinguished by the transverse HFK invariant, with or without naturality, because \(\widehat{\textit{HFK}}=0\) and HFK ⁻ has rank 1 in the relevant bidegree.

Note that [34], building on work from [33], uses an unusual convention for braids, so that a positive generator σ _k of the braid group is given topologically as a negative crossing in the usual knot theory sense. This has the effect of mirroring all topological knots and explains the μ ⁻¹ difference in conventions.

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Titel: A Topological Introduction to Knot Contact Homology
verfasst von: Lenhard Ng
Verlag: Springer International Publishing
Buch: Contact and Symplectic Topology
Print ISBN: 978-3-319-02035-8

Electronic ISBN: 978-3-319-02036-5

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-02036-5_10

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