Cellular Legendrian contact homology for surfaces, part II

Rutherford, Dan; Sullivan, Michael C.

doi:10.1142/s0129167x19500368

Cited by 14 publications

(50 citation statements)

References 33 publications

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“…Recall that it is proved by Rutherford–Sullivan in that the cellular dg algebra

C (Λ)

is quasi‐isomorphic to its Chekanov–Eliashberg algebra

C E (Λ)

defined over

K = Z / 2

. In particular, we have a quasi‐isomorphism

C false(Λ_{p, q, r} false) ≅ C E false(Λ_{p, q, r} false) .

Note that this quasi‐isomorphism preserves the

double-struckk

‐bimodule structures on both sides.…”

Section: Combinatorial Computationsmentioning

confidence: 88%

“…Note that one can recover from the formula in the cellular dg algebra

C ({normalΛ}_{p, q, r})

. In fact, this follows from the relations

b_{L}^{m, n} = b_{R}^{m, n} = b_{D}^{m, n} = 0,

a_{+, +}^{m, n} = a_{-, -}^{m, n} .

in the cellular dg algebra

{scriptC}_{| |} false(Λ_{p, q, r} false)

associated to the cellular decomposition

E_{false| false|}

obtained by shifting

p_{x} false(normalΣ false)

into the borders of the elementary squares in the transverse square decomposition

E_{⋔}

(see [, Section 3.6], Figure 12). In particular, the crossing arc corresponding to the 1‐cell

e_{1}^{1}

is shifted into the edges of

□_{7}

.…”

Section: Orientationsmentioning

confidence: 94%

“…Recall that in , the cellular dg algebra

C (Λ)

Λ \subset J^{1} false(S false)

is shown to be quasi‐isomorphic to the Chekanov–Eliashberg dg algebra

C E (Λ)

over

Z / 2

by fixing a transverse square decomposition

E_{⋔}

S

and doing local analysis of Morse flow trees for each elementary square. In our specific case when

Λ = {normalΛ}_{p, q, r}

with

p, q, r ⩾ 2

, it is not hard to see from Figure that one can find a transverse square decomposition

E_{⋔}

for

Λ_{p, q, r}

such that only Type I, II, III, IV and IX squares appear in

E_{⋔}

as elementary squares (see Figure ).…”

Section: Orientationsmentioning

confidence: 99%

“…By abuse of notations, we shall denote these critical points by

a_{\pm, \pm}^{m, n}, b_{U}^{m, n}, b_{R}^{m, n}, b_{D}^{m, n}, b_{L}^{m, n}, c_{7}^{m, n},

where

a_{\pm, \pm}^{m, n}

are the four corners of

□_{7}

, and they are minima of

f_{m n}

b_{U}^{m, n}, b_{R}^{m, n}, b_{D}^{m, n}, b_{L}^{m, n}

are the saddle points of

f_{m n}

located on the edges of

□_{7}

, and

c_{7}^{m, n}

is a local maximum of

f_{m n}

lying in the interior of

□_{7}

. By the analysis of , except for the

- \nabla f_{m, n}

flow lines from

c_{7}^{m, n}

b_{U}^{m, n}, b_{R}^{m, n}, b_{D}^{m, n}

and

b_{L}^{m, n}

, there are four additional rigid Morse flow trees with positive puncture at

c_{7}^{m, n}

(see Figure ). The first two Morse flow trees have a 2‐valent negative puncture at

c_{7}^{k, n}

and

c_{7}^{}

…”

Section: Orientationsmentioning

confidence: 99%

“…In order to simplify the analysis of the rigid Morse flow trees in

U (e_{α}^{1})

, one needs to choose carefully the local defining functions

f_{m}

of the sheets

S_{m} \subset {normalΛ}_{p, q, r}

(see [, Section 11] for details). For any point

false(x_{1}, x_{2} false) \in γ_{i}

such that

f_{m n} false(x_{1}, x_{2} false) > 0

, we require that

- \nabla f_{m n} false(x_{1}, x_{2} false)

is transverse to

γ_{i}

(x_{1}, x_{2})

and is inward pointing, so that the Reeb chords in

U (e_{α}^{1})

together with their differentials form a dg subalgebra of

C E ({normalΛ}_{p, q, r})

.…”

Section: Orientationsmentioning

confidence: 99%

See 4 more Smart Citations

Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Let M be a Liouville 6‐manifold which is the smooth fiber of a Lefschetz fibration on C4 constructed by suspending a Lefschetz fibration on C3. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category F(M) and the wrapped Fukaya category W(M) are related through A∞‐Koszul duality, by identifying them with cyclic and Calabi–Yau completions of the same quiver algebra. This implies the split‐generation of the compact Fukaya category F(M) by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi‐dilations in the sense of Seidel–Solomon are obtained.

show abstract

“…Recall that it is proved by Rutherford–Sullivan in that the cellular dg algebra

C (Λ)

is quasi‐isomorphic to its Chekanov–Eliashberg algebra

C E (Λ)

defined over

K = Z / 2

. In particular, we have a quasi‐isomorphism

C false(Λ_{p, q, r} false) ≅ C E false(Λ_{p, q, r} false) .

Note that this quasi‐isomorphism preserves the

double-struckk

‐bimodule structures on both sides.…”

Section: Combinatorial Computationsmentioning

confidence: 88%

“…Note that one can recover from the formula in the cellular dg algebra

C ({normalΛ}_{p, q, r})

. In fact, this follows from the relations

b_{L}^{m, n} = b_{R}^{m, n} = b_{D}^{m, n} = 0,

a_{+, +}^{m, n} = a_{-, -}^{m, n} .

in the cellular dg algebra

{scriptC}_{| |} false(Λ_{p, q, r} false)

associated to the cellular decomposition

E_{false| false|}

obtained by shifting

p_{x} false(normalΣ false)

into the borders of the elementary squares in the transverse square decomposition

E_{⋔}

(see [, Section 3.6], Figure 12). In particular, the crossing arc corresponding to the 1‐cell

e_{1}^{1}

is shifted into the edges of

□_{7}

.…”

Section: Orientationsmentioning

confidence: 94%

“…Recall that in , the cellular dg algebra

C (Λ)

Λ \subset J^{1} false(S false)

is shown to be quasi‐isomorphic to the Chekanov–Eliashberg dg algebra

C E (Λ)

over

Z / 2

by fixing a transverse square decomposition

E_{⋔}

S

and doing local analysis of Morse flow trees for each elementary square. In our specific case when

Λ = {normalΛ}_{p, q, r}

with

p, q, r ⩾ 2

, it is not hard to see from Figure that one can find a transverse square decomposition

E_{⋔}

for

Λ_{p, q, r}

such that only Type I, II, III, IV and IX squares appear in

E_{⋔}

as elementary squares (see Figure ).…”

Section: Orientationsmentioning

confidence: 99%

“…By abuse of notations, we shall denote these critical points by

a_{\pm, \pm}^{m, n}, b_{U}^{m, n}, b_{R}^{m, n}, b_{D}^{m, n}, b_{L}^{m, n}, c_{7}^{m, n},

where

a_{\pm, \pm}^{m, n}

are the four corners of

□_{7}

, and they are minima of

f_{m n}

b_{U}^{m, n}, b_{R}^{m, n}, b_{D}^{m, n}, b_{L}^{m, n}

are the saddle points of

f_{m n}

located on the edges of

□_{7}

, and

c_{7}^{m, n}

is a local maximum of

f_{m n}

lying in the interior of

□_{7}

. By the analysis of , except for the

- \nabla f_{m, n}

flow lines from

c_{7}^{m, n}

b_{U}^{m, n}, b_{R}^{m, n}, b_{D}^{m, n}

and

b_{L}^{m, n}

, there are four additional rigid Morse flow trees with positive puncture at

c_{7}^{m, n}

(see Figure ). The first two Morse flow trees have a 2‐valent negative puncture at

c_{7}^{k, n}

and

c_{7}^{}

…”

Section: Orientationsmentioning

confidence: 99%

“…In order to simplify the analysis of the rigid Morse flow trees in

U (e_{α}^{1})

, one needs to choose carefully the local defining functions

f_{m}

of the sheets

S_{m} \subset {normalΛ}_{p, q, r}

(see [, Section 11] for details). For any point

false(x_{1}, x_{2} false) \in γ_{i}

such that

f_{m n} false(x_{1}, x_{2} false) > 0

, we require that

- \nabla f_{m n} false(x_{1}, x_{2} false)

is transverse to

γ_{i}

(x_{1}, x_{2})

and is inward pointing, so that the Reeb chords in

U (e_{α}^{1})

together with their differentials form a dg subalgebra of

C E ({normalΛ}_{p, q, r})

.…”

Section: Orientationsmentioning

confidence: 99%

See 3 more Smart Citations

Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Sheaves via augmentations of Legendrian surfaces

Rutherford

Sullivan

2021

J. Homotopy Relat. Struct.

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Augmentations and immersed Lagrangian fillings

Pan

Rutherford

2023

Journal of Topology

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For a Legendrian link Λ ⊂ 𝐽 1 𝑀 with 𝑀 = ℝ or 𝑆 1 , immersed exact Lagrangian fillings 𝐿 ⊂ Symp(𝐽 1 𝑀) ≅ 𝑇 * (ℝ >0 × 𝑀) of Λ can be lifted to conical Legendrian fillings Σ ⊂ 𝐽 1 (ℝ >0 × 𝑀) of Λ. When Σ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom. 19 (2021), no. 3, 635-722], for each augmentation 𝛼 ∶ (Σ) → ℤ∕2 of the LCH algebra of Σ, there is an induced augmentation 𝜖 (Σ,𝛼) ∶ (Λ) → ℤ∕2. With Σ fixed, the set of homotopy classes of all such induced augmentations, 𝐼 Σ ⊂ 𝐴𝑢𝑔(Λ)∕∼, is a Legendrian isotopy invariant of Σ. We establish methods to compute 𝐼 Σ based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math. 374 (2020), 107348, 71 pp.] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary 𝑛 ⩾ 1, we give examples of Legendrian torus knots with 2𝑛 distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when 𝜌 ≠ 1 and Λ ⊂ 𝐽 1 ℝ, every 𝜌-graded augmentation of Λ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of 𝜌-graded augmented Legendrian cobordism.M S C 2 0 2 0 53D42 (primary)

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Cellular Legendrian contact homology for surfaces, part II

Cited by 14 publications

References 33 publications

Koszul duality via suspending Lefschetz fibrations

Koszul duality via suspending Lefschetz fibrations

Sheaves via augmentations of Legendrian surfaces

Augmentations and immersed Lagrangian fillings

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