Torus Knots and the Topological Vertex

Jockers, Hans; Klemm, Albrecht; Soroush, Masoud

doi:10.1007/s11005-014-0687-0

Cited by 14 publications

(34 citation statements)

References 68 publications

(221 reference statements)

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“…For torus knots, the corresponding open Gromov-Witten invariants of (X, L K ) have been computed directly using localization in [10], and using mirror symmetry in [20,21]. However, the mirror symmetry approach of [20] does not readily generalize to non-toric knots.…”

Section: Open Gromov-witten Invariants and The Augmentation Polynomialmentioning

confidence: 99%

“…In addition to the apparent coincidence between the unknot's augmentation polynomial and the mirrorX, there are also physical arguments for this conjecture coming from the connections between topological string theory, Chern-Simons theory, and the HOMFLY polynomial [11,12,21,24] …”

Section: Open Gromov-witten Invariants and The Augmentation Polynomialmentioning

confidence: 99%

“…are the maximum degrees [21]. Fortunately, to compute K d,w for a given d and w, only a series solution for p is needed.…”

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%

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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“…9 The wave function corresponding to V Hopf (2) is obtained by taking a contour C (2) which is a small circle p 2 = x 1 ; this gives…”

Section: An Example: Hopf Linkmentioning

confidence: 99%

“…. , x n ), where by assumption these strings are massive and we have integrated them out to 9 We have simplified things slightly by shifts of variables. For details see Appendix A. get ∆(x).…”

Section: Knot Parallels and Higher Rank Representationsmentioning

confidence: 99%

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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Sequencing BPS spectra

Gukov

Nawata

Saberi

et al. 2016

J. High Energ. Phys.

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This paper provides both a detailed study of color-dependence of link homologies, as realized in physics as certain spaces of BPS states, and a broad study of the behavior of BPS states in general. We consider how the spectrum of BPS states varies as continuous parameters of a theory are perturbed. This question can be posed in a wide variety of physical contexts, and we answer it by proposing that the relationship between unperturbed and perturbed BPS spectra is described by a spectral sequence. These general considerations unify previous applications of spectral sequence techniques to physics, and explain from a physical standpoint the appearance of many spectral sequences relating various link homology theories to one another. We also study structural properties of colored HOMFLY homology for links and evaluate Poincaré polynomials in numerous examples. Among these structural properties is a novel "sliding" property, which can be explained by using (refined) modular S-matrix. This leads to the identification of modular transformations in Chern-Simons theory and 3d N = 2 theory via the 3d/3d correspondence. Lastly, we introduce the notion of associated varieties as classical limits of recursion relations of colored superpolynomials of links, and study their properties.

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Torus Knots and the Topological Vertex

Cited by 14 publications

References 68 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

Sequencing BPS spectra

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