A Topological Introduction to Knot Contact Homology

Ng, Lenhard

doi:10.1007/978-3-319-02036-5_10

Cited by 29 publications

(67 citation statements)

References 44 publications

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“…The Aganagic-Vafa conjecture can be motivated by observing that the mirror toX given in Equation (5) can be written asX = {uv = A(x, p; Q)} , where A(x, p; Q) = 1 − Qx − Qp + Qxp is the augmentation polynomial of the unknot in Legendrian contact homology [13,22]. The moduli of Lagrangian submanifolds with topology S 1 × R 2 was described by the zero locus {A(x, p; Q) = 0} ⊂X.…”

Section: Open Gromov-witten Invariants and The Augmentation Polynomialmentioning

confidence: 99%

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Open Gromov-Witten Invariants from the Augmentation Polynomial

Mahowald

2017

Symmetry

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A conjecture of Aganagic and Vafa relates the open Gromov-Witten theory of X = O P 1 (−1, −1) to the augmentation polynomial of Legendrian contact homology. We describe how to use this conjecture to compute genus zero, one boundary component open Gromov-Witten invariants for Lagrangian submanifolds L K ⊂ X obtained from the conormal bundles of knots K ⊂ S 3 . This computation is then performed for two non-toric examples (the figure-eight and three-twist knots). For (r, s) torus knots, the open Gromov-Witten invariants can also be computed using Atiyah-Bott localization. Using this result for the unknot and the (3, 2) torus knot, we show that the augmentation polynomial can be derived from these open Gromov-Witten invariants.

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Section: Open Gromov-witten Invariants and The Augmentation Polynomialmentioning

confidence: 99%

“…The moduli space of such augmentations is described by an equation A K (x, p, Q) = 0, where x, p, and Q are generators for H 2 (U * S 3 , Λ K ). A K is called the augmentation polynomial of the knot K (more detailed accounts of Legendrian contact homology can be found in [12,13]). …”

Section: Introductionmentioning

confidence: 99%

Open Gromov-Witten Invariants from the Augmentation Polynomial

Mahowald

2017

Symmetry

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“…As a result, some of the discussion below is rather imprecise, but we will provide references to more detailed treatments in the literature for the interested reader. A slightly less brief overview can be found in [26].…”

Section: Mathematical Overview Of Knot Contact Homologymentioning

confidence: 99%

“…Given a braid with m strands such that K is the closure of the braid (i.e., the result of gluing together corresponding ends of the braid in S 3 ), there is a combinatorial formula for the differential graded algebra (A, ∂) associated to K. Outside of computations, we will not need the precise formula, and we omit it here; please see the Appendix of [26] for the full definition of the version of the invariant that we use in this paper, noting that our Q is denoted by U in that paper. (The algebra given in [26] is in turn based on work that originally appeared in the series of papers [30,31,32,33].)…”

Section: Mathematical Overview Of Knot Contact Homologymentioning

confidence: 99%

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Topological strings, D-model, and knot contact homology

Aganagic

Ekholm

et al. 2014

Advances in Theoretical and Mathematical Physics

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We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed Apolynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study arXiv:1304.5778v2 [hep-th] 23 Dec 2013 of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model." In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Q-deformation of the extension of A-polynomials associated with the link complement.

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Knot Invariants from Topological Recursion on Augmentation Varieties

Gu¹,

Jockers²,

Klemm³

et al. 2014

Commun. Math. Phys.

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Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on S3 and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on S 3 and their invariants encoded in colored HOMFLY polynomials by means of topological recursion. In the context of the local mirror Calabi-Yau threefold of the resolved conifold, we generalize the topological recursion of the remodelled B-model in order to study branes beyond the class of toric HarveyLawson special Lagrangians -as required for analyzing non-trivial knots on S 3 . The basic ingredients for the proposed recursion are the spectral curve, given by the augmentation variety of the knot, and the calibrated annulus kernel, encoding the topological annulus amplitudes associated to the knot. We present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion. We further argue that -if an explicit form of the calibrated annulus kernel is provided for any other knot -the proposed topological recursion should still be applicable. We study the implications of our proposal for knot theory, which exhibit interesting consequences for colored HOMFLY polynomials of mutant knots.

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A Topological Introduction to Knot Contact Homology

Abstract: This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.

Cited by 29 publications

References 44 publications

Open Gromov-Witten Invariants from the Augmentation Polynomial

Open Gromov-Witten Invariants from the Augmentation Polynomial

Topological strings, D-model, and knot contact homology

Knot Invariants from Topological Recursion on Augmentation Varieties

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