The correspondence between augmentations and rulings for Legendrian knots

Ng, Lenhard; Sabloff, Joshua M.

doi:10.2140/pjm.2006.224.141

Cited by 20 publications

(31 citation statements)

References 9 publications

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“…The knot contact homology describes knot invariants as invariants of the Legendrian submanifolds in the contact manifold [26,[45][46][47][48][49][50][51][52][53][54][55]. In particular, the geometric set-up of the knot contact homology in [56,57] is similar to the topological A-model on T * M [20][21][22], and the string-theoretical interpretation of the knot contact homology has been recently studied in [17].…”

Section: Appendix B Knot Contact Homology and Augmentation Polynomialmentioning

confidence: 99%

Super-A-polynomial for knots and BPS states

2013

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We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the "refined A-polynomial" introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for S r -colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d N = 2 theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot.

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Section: Appendix B Knot Contact Homology and Augmentation Polynomialmentioning

confidence: 99%

Super-A-polynomial for knots and BPS states

2013

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“…Ng and Sabloff also worked out a surjective correspondence [19] that assigns a Z-graded ruling to each augmentation. In that correspondence, the size of the preimage of each Z-graded ruling ρ of the front diagram f is the number 2 In particular, the number of augmentations belonging to ρ depends on θ(ρ) and the diagram only.…”

Section: Rulingsmentioning

confidence: 99%

Braid-positive Legendrian links

Kálmán

2006

International Mathematics Research Notices

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Any link that is the closure of a positive braid has a natural Legendrian representative. These were introduced in [15], where their ChekanovEliashberg contact homology was also evaluated. In this paper we rephrase and improve that computation using a matrix representation. In particular, we present a way of finding all augmentations of such Legendrians, construct an augmentation which is also a ruling, and find surprising links to LU -decompositions and Gröbner bases.

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“…From the algorithmic construction of the map Ψ : Aug(D) → R 0 (D) in [16], Ψ(ǫ i ) is ρ for all 1 ≤ i ≤ m. From Theorem 7.3 of [10], P ǫ i (t) = P C i (t) holds. Consequently, we may use the chord path approach given in Section 2.3 to compute P ǫ i (t).…”

Section: The Main Resultsmentioning

confidence: 99%

“…Fuchs and Ishkanov [8] and, independently, Sabloff [18] prove the converse. Ng and Sabloff [16] further clarify the relationship between augmentations and graded normal rulings by proving that there exists an algorithmically defined many-to-one map Ψ : Aug(D) → R 0 (D). Josh Sabloff posed the following question to the second author, "Does Ψ(ǫ 1 ) = Ψ(ǫ 2 ) imply P ǫ 1 (t) = P ǫ 2 (t)?"…”

Section: Introductionmentioning

confidence: 98%

Computing homology invariants of Legendrian knots

Casey

Henry

2014

J. Knot Theory Ramifications

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Abstract. The Chekanov-Eliashberg differential graded algebra of a Legendrian knot L is a rich source of Legendrian knot invariants, as is the theory of generating families. The set P (L) of homology groups of augmentations of the Chekanov-Eliashberg algebra is an invariant, as is a count of objects from the theory of generating families called graded normal rulings. This article gives two results demonstrating the usefulness of computing the homology group of an augmentation using a combinatorial interpretation of a generating family called a Morse complex sequence [10]. First, we show that if the projection of L to the xz-plane has exactly 4 cusps, then |P (L)| ≤ 1. Second, we show that two augmentations associated to the same graded normal ruling by the many-to-one map between augmentations and graded normal rulings defined by Ng and Sabloff [16] need not have isomorphic homology groups.

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The correspondence between augmentations and rulings for Legendrian knots

Abstract: We strengthen the link between holomorphic and generating-function invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot's contact homology to the complete ruling invariant of Chekanov and Pushkar.

Cited by 20 publications

References 9 publications

Super-A-polynomial for knots and BPS states

Super-A-polynomial for knots and BPS states

Braid-positive Legendrian links

Computing homology invariants of Legendrian knots

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