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arxiv: 2401.13024 · v2 · pith:JYQRADK7new · submitted 2024-01-23 · 🧮 math.SG

Augmentation varieties and disk potentials III

Kenneth Blakey , Soham Chanda , Yuhan Sun , Chris T. Woodward This is my paper

Pith reviewed 2026-05-24 04:18 UTC · model grok-4.3

classification 🧮 math.SG
keywords augmentation varietydisk potentialLegendrian coversmonotone Lagrangian toriChekanov-Eliashberg algebracircle-fibered contact manifoldsexact fillingsLegendrian isotopy
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The pith

For connected Legendrian covers of monotone Lagrangian tori, the augmentation variety equals the image of the zero level set of the disk potential.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an equality between the augmentation variety and the image of the zero level set of the disk potential for connected Legendrian covers of monotone Lagrangian tori in circle-fibered contact manifolds. The result confirms a suggestion by Dimitroglou-Rizell-Golovko and applies it to show that Legendrian lifts of Vianna's exotic tori are not isotopic. It further demonstrates that the Legendrian lift of the Clifford torus has no exact Lagrangian fillings. The work also addresses disconnected Legendrians by linking components of the augmentation variety to partitions via Lagrangian fillings.

Core claim

The paper proves that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential.

What carries the argument

The augmentation variety of the Chekanov-Eliashberg algebra, set equal to the image of the zero level set of the disk potential.

If this is right

  • Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic.
  • The Legendrian lift of the Clifford torus admits no exact fillings.
  • For certain disconnected Legendrians, the components of the augmentation variety correspond to partitions, each defined by a Lagrangian filling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This equality may allow computation of one invariant from the other in related geometric settings.
  • Similar relations could hold for non-monotone or higher-genus Lagrangians.
  • The approach might extend to other contact manifolds beyond circle-fibered ones.

Load-bearing premise

The Legendrians considered are connected covers of monotone Lagrangian tori in circle-fibered contact manifolds.

What would settle it

A counterexample consisting of a connected Legendrian cover of a monotone Lagrangian torus where the augmentation variety does not match the image of the disk potential zero set would disprove the claim.

Figures

Figures reproduced from arXiv: 2401.13024 by Chris T. Woodward, Kenneth Blakey, Soham Chanda, Yuhan Sun.

Figure 1
Figure 1. Figure 1: Basis to ensure non-negative exponents We now prove the relation to the zero level set of the disk potential. Let Z be a negative circle bundle over a monotone symplectic manifold Y with minimal Chern number at least two. Suppose Λ ⊂ Z a connected Legendrian lift of a connected compact monotone Lagrangian Π with minimal Maslov number two and equipped with a relative spin structure. Let WΠ : Rep(Π) → C be t… view at source ↗
Figure 2
Figure 2. Figure 2: Newton polytope for two and three-dimensional tori cor￾responding to to (a, b, c). Here f, g are non-constant polynomials. Then, from [38] (or Lemma 2.1 in [26]) we have Newt(WΛabc d ) = Newt(f) + Newt(g), where ‘+’ denotes Minkowski sum. By setting y = 0, we get a factorization of WΛabc d−1 , which we know is irreducible. This implies either f or g is a polynomial solely consisting of the variable y. With… view at source ↗
Figure 3
Figure 3. Figure 3: Homological relations between components of moduli spaces M1(Λ, A, k)dim(A)+1 → M(Λ, A, k)dim(A) . Each fiber evaluates to the translation of the boundary ∂uk by an element of A, the moduli space being invariant under multiplication by A. It follows that the homology class of the image in Λ = ∂L is [ϕΛ,k(Mc1(Λ, A, k)] = µk[A] ∈ Hn−2(L, Z2) [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
read the original abstract

This is the third in a series of papers in which we construct Chekanov-Eliashberg algebras for Legendrians in circle-fibered contact manifolds and study the associated augmentation varieties. In this part, we prove that for connected Legendrian covers of monotone Lagrangian tori, the augmentation variety is equal to the image of the zero level set of the disk potential, as suggested by Dimitroglou-Rizell-Golovko. In particular, we show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic. Using related ideas, we show that the Legendrian lift of the Clifford torus admits no exact fillings, extending results of Dimitroglou-Rizell and Treumann-Zaslow in dimension two. We consider certain disconnected Legendrians, and show, similar to another suggestion of Aganagic-Ekholm-Ng-Vafa that the components of the augmentation variety correspond to certain partitions and each component is defined by a (not necessarily exact) Lagrangian filling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript, the third in a series, proves that for connected Legendrian covers of monotone Lagrangian tori in circle-fibered contact manifolds the augmentation variety equals the image of the zero level set of the disk potential, confirming a suggestion of Dimitroglou-Rizell-Golovko. It applies the result to show that Legendrian lifts of Vianna's exotic tori are not Legendrian isotopic and that the Legendrian lift of the Clifford torus admits no exact fillings. For certain disconnected Legendrians it shows that components of the augmentation variety correspond to partitions, each defined by a (not necessarily exact) Lagrangian filling.

Significance. If the central equality holds, the work supplies a concrete geometric realization of the augmentation variety in terms of the disk potential, enabling direct applications to Legendrian isotopy questions and filling obstructions. The results on Vianna tori and the Clifford torus extend dimension-two phenomena to higher dimensions, while the disconnected case aligns with an independent suggestion of Aganagic-Ekholm-Ng-Vafa. The cumulative construction across the series is a strength when the dependencies are clearly tracked.

minor comments (3)
  1. [Abstract] Abstract: the main theorem statement would be clearer if it explicitly named the ambient dimension or the precise class of circle-fibered contact manifolds in which the equality is proved.
  2. [Introduction] Introduction: the proof of the central equality relies on constructions from Parts I and II; specific theorem or proposition numbers from those papers should be cited when the key definitions or comparison maps are invoked.
  3. [Disconnected Legendrians] Section treating disconnected Legendrians: the claimed correspondence between augmentation-variety components and partitions is stated, but an explicit low-dimensional example showing how a non-exact filling defines one component would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its significance in realizing the augmentation variety geometrically via the disk potential, and the recommendation for minor revision. We are pleased that the connections to the suggestions of Dimitroglou-Rizell-Golovko and Aganagic-Ekholm-Ng-Vafa are noted, as well as the extensions of dimension-two results.

Circularity Check

0 steps flagged

No significant circularity; central equality established by direct proof

full rationale

The paper's main result is a theorem proving equality between the augmentation variety (from the Chekanov-Eliashberg DGA) and the image of the zero level set of the disk potential for connected Legendrian covers of monotone Lagrangian tori. This is achieved via explicit construction and comparison in the present work, extending prior suggestions from non-overlapping authors (Dimitroglou-Rizell-Golovko et al.). Although the paper is part III of a series, the load-bearing step is the proof itself rather than a self-referential definition, fitted parameter renamed as prediction, or uniqueness theorem imported from the authors' own prior work. No equations or claims reduce by construction to inputs; the result is externally falsifiable via Legendrian isotopy and filling questions. This is the normal case of a self-contained geometric proof.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on the circle-fibered contact manifold setting, monotonicity of the tori, and constructions from the preceding papers in the series plus suggestions from Dimitroglou-Rizell-Golovko and Aganagic-Ekholm-Ng-Vafa.

axioms (2)
  • domain assumption Contact manifolds are circle-fibered
    Explicitly stated as the ambient setting in the abstract.
  • domain assumption Legendrians are connected covers of monotone Lagrangian tori
    Required for the main equality statement.

pith-pipeline@v0.9.0 · 5705 in / 1267 out tokens · 21013 ms · 2026-05-24T04:18:33.663415+00:00 · methodology

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages · 2 internal anchors

  1. [1]

    Abraham, J

    R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, tensor analysis, and applications, volume 75 of Applied Mathematical Sciences . Springer-Verlag, New York, second edition, 1988

    work page 1988

  2. [2]

    Topologi- cal strings, D-model, and knot contact homology

    Mina Aganagic, Tobias Ekholm, Lenhard Ng, and Cumrun Vafa. Topologi- cal strings, D-model, and knot contact homology. Adv. Theor. Math. Phys. , 18(4):827–956, 2014

    work page 2014

  3. [3]

    Mirror Symmetry, D-Branes and Counting Holomorphic Discs

    Mina Aganagic and Cumrun Vafa. Mirror symmetry, d-branes and counting holomorphic discs, 2000. https://arxiv.org/abs/hep-th/0012041

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  4. [4]

    Augmen- tation varieties and disk potentials I

    Kenneth Blakey, Soham Chanda, Yuhan Sun, and Chris Woodward. Augmen- tation varieties and disk potentials I

    work page

  5. [5]

    Augmen- tation varieties and disk potentials II

    Kenneth Blakey, Soham Chanda, Yuhan Sun, and Chris Woodward. Augmen- tation varieties and disk potentials II

    work page

  6. [6]

    Vector fields and characteristic numbers

    Raoul Bott. Vector fields and characteristic numbers. Michigan Math. J. , 14:231–244, 1967

    work page 1967

  7. [7]

    A survey of contact homology

    Fr´ ed´ eric Bourgeois. A survey of contact homology. In New perspectives and challenges in symplectic field theory , volume 49 of CRM Proc. Lecture Notes , pages 45–71. Amer. Math. Soc., Providence, RI, 2009

    work page 2009

  8. [8]

    Bilinearized Legendrian contact homology and the augmentation category

    Fr´ ed´ eric Bourgeois and Baptiste Chantraine. Bilinearized Legendrian contact homology and the augmentation category. J. Symplectic Geom., 12(3):553–583, 2014

    work page 2014

  9. [9]

    Infinitely many monotone lagrangian tori in higher projective spaces

    Soham Chanda, Amanda Hirschi, and Luya Wang. Infinitely many monotone lagrangian tori in higher projective spaces. arXiv preprint arXiv:2307.06934 , 2023

    work page arXiv 2023

  10. [10]

    Floer homology and Lagrangian concordance

    Baptiste Chantraine, Georgios Dimitroglou Rizell, Paolo Ghiggini, and Roman Golovko. Floer homology and Lagrangian concordance. In Proceedings of the G¨ okova Geometry-Topology Conference 2014, pages 76–113. G¨ okova Geome- try/Topology Conference (GGT), G¨ okova, 2015. AUGMENTATION VARIETIES AND DISK POTENTIALS 43

    work page 2014

  11. [11]

    Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds

    Cheol-Hyun Cho and Yong-Geun Oh. Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math., 10(4):773–814, 2006

    work page 2006

  12. [12]

    Symplectic hypersurfaces and transversality in Gromov-Witten theory

    Kai Cieliebak and Klaus Mohnke. Symplectic hypersurfaces and transversality in Gromov-Witten theory. J. Symplectic Geom., 5(3):281–356, 2007

    work page 2007

  13. [13]

    Obstructions to La- grangian concordance

    Christopher Cornwell, Lenhard Ng, and Steven Sivek. Obstructions to La- grangian concordance. Algebr. Geom. Topol., 16(2):797–824, 2016

    work page 2016

  14. [14]

    Knotted Legendrian surfaces with few Reeb chords

    Georgios Dimitroglou Rizell. Knotted Legendrian surfaces with few Reeb chords. Algebr. Geom. Topol., 11(5):2903–2936, 2011

    work page 2011

  15. [15]

    Lifting pseudo-holomorphic polygons to the sym- plectisation of P × R and applications

    Georgios Dimitroglou Rizell. Lifting pseudo-holomorphic polygons to the sym- plectisation of P × R and applications. Quantum Topol., 7(1):29–105, 2016

    work page 2016

  16. [16]

    Legendrian submanifolds from bohr–sommerfeld covers of monotone lagrangian tori

    Georgios Dimitroglou Rizell and Roman Golovko. Legendrian submanifolds from bohr–sommerfeld covers of monotone lagrangian tori. To appear in Comm. Anal. Geom

    work page

  17. [17]

    On homological rigid- ity and flexibility of exact Lagrangian endocobordisms

    Georgios Dimitroglou Rizell and Roman Golovko. On homological rigid- ity and flexibility of exact Lagrangian endocobordisms. Internat. J. Math. , 25(10):1450098, 24, 2014

    work page 2014

  18. [18]

    Augmentations, annuli, and alexander polyno- mials, 2020

    Lu´ ıs Diogo and Tobias Ekholm. Augmentations, annuli, and alexander polyno- mials, 2020. https://arxiv.org/abs/2005.09733

    work page arXiv 2020

  19. [19]

    Legendrian contact homol- ogy in P × R

    Tobias Ekholm, John Etnyre, and Michael Sullivan. Legendrian contact homol- ogy in P × R. Trans. Amer. Math. Soc., 359(7):3301–3335, 2007

    work page 2007

  20. [20]

    Legendrian knots and exact Lagrangian cobordisms

    Tobias Ekholm, Ko Honda, and Tam´ as K´ alm´ an. Legendrian knots and exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) , 18(11):2627–2689, 2016

    work page 2016

  21. [21]

    Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials

    Tobias Ekholm and Lenhard Ng. Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials. Adv. Theor. Math. Phys. , 24(8):2067–2145, 2020

    work page 2067

  22. [22]

    Nonfillable Legendrian knots in the 3-sphere.Algebr

    Tolga Etg¨ u. Nonfillable Legendrian knots in the 3-sphere.Algebr. Geom. Topol., 18(2):1077–1088, 2018

    work page 2018

  23. [23]

    Lagrangian inter- section Floer theory: anomaly and obstruction

    Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian inter- section Floer theory: anomaly and obstruction. Part I , volume 46 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009

    work page 2009

  24. [24]

    Lagrangian inter- section Floer theory: anomaly and obstruction

    Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian inter- section Floer theory: anomaly and obstruction. Part II , volume 46 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009

    work page 2009

  25. [25]

    Augmentations, fillings, and clusters, 2020

    Honghao Gao, Linhui Shen, and Daping Weng. Augmentations, fillings, and clusters, 2020. https://arxiv.org/abs/2008.10793

    work page arXiv 2020

  26. [26]

    Absolute irreducibility of polynomials via newton polytopes

    Shuhong Gao. Absolute irreducibility of polynomials via newton polytopes. Journal of Algebra, 237(2):501–520, 2001

    work page 2001

  27. [27]

    R. Golovko. A note on the front spinning construction. Bull. Lond. Math. Soc. , 46(2):258–268, 2014

    work page 2014

  28. [28]

    A note on the infinite number of exact Lagrangian fillings for spherical spuns

    Roman Golovko. A note on the infinite number of exact Lagrangian fillings for spherical spuns. Pacific J. Math. , 317(1):143–152, 2022. AUGMENTATION VARIETIES AND DISK POTENTIALS 44

    work page 2022

  29. [29]

    Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc

    Sheldon Katz and Chiu-Chu Melissa Liu. Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. In The interaction of finite-type and Gromov-Witten invariants (BIRS 2003), volume 8 of Geom. Topol. Monogr., pages 1–47. Geom. Topol. Publ., Coventry, 2006

    work page 2003

  30. [30]

    Product structures in Floer theory for Lagrangian cobordisms

    No´ emie Legout. Product structures in Floer theory for Lagrangian cobordisms. J. Symplectic Geom., 18(6):1647–1750, 2020

    work page 2020

  31. [31]

    Open Gromov-Witten invariants on elliptic K3 surfaces and wall- crossing

    Yu-Shen Lin. Open Gromov-Witten invariants on elliptic K3 surfaces and wall- crossing. Comm. Math. Phys. , 349(1):109–164, 2017

    work page 2017

  32. [32]

    David S. Metzler. Cohomological localization for manifolds with boundary. Int. Math. Res. Not. , (24):1239–1274, 2002

    work page 2002

  33. [33]

    Framed knot contact homology

    Lenhard Ng. Framed knot contact homology. Duke Math. J. , 141(2):365–406, 2008

    work page 2008

  34. [34]

    The cardinality of the augmentation category of a Legendrian link.Math

    Lenhard Ng, Dan Rutherford, Vivek Shende, and Steven Sivek. The cardinality of the augmentation category of a Legendrian link.Math. Res. Lett., 24(6):1845– 1874, 2017

    work page 2017

  35. [35]

    Riemann-Hilbert problem and application to the perturbation theory of analytic discs

    Yong-Geun Oh. Riemann-Hilbert problem and application to the perturbation theory of analytic discs. Kyungpook Math. J., 35(1):39–75, 1995

    work page 1995

  36. [37]

    Knot invariants and topological strings

    Hirosi Ooguri and Cumrun Vafa. Knot invariants and topological strings. Nu- clear Phys. B , 577(3):419–438, 2000

    work page 2000

  37. [38]

    On multiplication and factorization of polynomials, I

    Alexander M Ostrowski. On multiplication and factorization of polynomials, I. Lexicographic orderings and extreme aggregates of terms. aequationes mathe- maticae, 13(3):201–228, 1975

    work page 1975

  38. [39]

    Exact Lagrangian fillings of Legendrian (2 , n) torus links

    Yu Pan. Exact Lagrangian fillings of Legendrian (2 , n) torus links. Pacific J. Math., 289(2):417–441, 2017

    work page 2017

  39. [40]

    Pandharipande, J

    R. Pandharipande, J. Solomon, and J. Walcher. Disk enumeration on the quintic 3-fold. J. Amer. Math. Soc. , 21(4):1169–1209, 2008

    work page 2008

  40. [41]

    Contact homology and virtual fundamental cycles

    John Pardon. Contact homology and virtual fundamental cycles. J. Amer. Math. Soc., 32(3):825–919, 2019

    work page 2019

  41. [42]

    The wall-crossing formula and La- grangian mutations

    James Pascaleff and Dmitry Tonkonog. The wall-crossing formula and La- grangian mutations. Adv. Math., 361:106850, 67, 2020

    work page 2020

  42. [43]

    Sabloff and Lisa Traynor

    Joshua M. Sabloff and Lisa Traynor. Obstructions to Lagrangian cobordisms between Legendrians via generating families. Algebr. Geom. Topol., 13(5):2733– 2797, 2013

    work page 2013

  43. [44]

    Cubic planar graphs and Legendrian surface theory

    David Treumann and Eric Zaslow. Cubic planar graphs and Legendrian surface theory. Adv. Theor. Math. Phys. , 22(5):1289–1345, 2018

    work page 2018

  44. [45]

    Infinitely many exotic monotone Lagrangian tori in CP2

    Renato Ferreira de Velloso Vianna. Infinitely many exotic monotone Lagrangian tori in CP2. J. Topol., 9(2):535–551, 2016

    work page 2016

  45. [46]

    Orientations for pseudoholomorphic quilts

    Katrin Wehrheim and Chris Woodward. Orientations for pseudoholomorphic quilts, 2015. https://arxiv.org/abs/1503.07803

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  46. [47]

    Floer field theory for coprime rank and degree

    Katrin Wehrheim and Chris Woodward. Floer field theory for coprime rank and degree. Indiana Univ. Math. J. , 69(6):2035–2088, 2020

    work page 2035

  47. [48]

    Pseudocycles and integral homology

    Aleksey Zinger. Pseudocycles and integral homology. Trans. Amer. Math. Soc., 360(5):2741–2765, 2008. AUGMENTATION VARIETIES AND DISK POTENTIALS 45 Department of Mathematics,182 Memorial Drive, Cambridge, MA 02139, U.S.A. Email address : kblakey@mit.edu Mathematics-Hill Center, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, U.S.A. E...

    work page 2008