---
title: Linearized Legendrian Contact Homology
url: https://www.emergentmind.com/topics/linearized-legendrian-contact-homology-lch
type: topic
---

# Linearized Legendrian Contact Homology

family \(\Lambda_k\) (for \(k\geq 1\)), where each \(\Lambda_k\) is a specific Legendrian knot (e.g., \(\Lambda_1\) is a \(m(9_{45})\) knot). For these knots, using explicit DGAs and augmentations given by 

\[ \epsilon_n(a_1) = n, \quad \epsilon_n(a_2) = -1, \quad \epsilon_n(a_3) = 1 \]

(properly extended), one finds that 

\[ LCH_*^{\epsilon_n}(\Lambda_k) \cong \begin{cases} 
\mathbb{Z} & *=1 \\
\mathbb{Z}^2 & *=0 \\
\mathbb{Z}/n & *=k\text{ or }*-k-1 \,(k>1) \\
0 & \text{otherwise} 
\end{cases} \]

For \(k=1\), torsion appears in degree \(-2\). 

**Interpretation:** By adjusting \(n\), one can generate torsion of arbitrary order.

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## 3. **Implications for *Geometric* Augmentations and Lagrangian Fillings**

### a. Seidel Isomorphism and Geometric Augmentations

If an augmentation is **geometric** (i.e., it is induced by an exact Lagrangian filling \(L\) via a local system), then by a (mod 2 graded) Seidel isomorphism (as in works by Gao–Rutherford and Ekholm et al.), 

\[ LCH_*^\epsilon(\Lambda;k) \cong H_{*+1}(L, \Lambda; k) \]

for \(k\) a field, and similarly for \(\mathbb{Z}\)-coefficients. 

Therefore, the ranks of LCH are constrained (and in particular, cannot have arbitrarily large torsion).

### b. Torsion Not Realizable by Fillings

**Main Consequence:** 

There exist \(\mathbb{Z}\)-valued augmentations of Legendrian knots (such as those above) whose associated LCH over \(\mathbb{Z}\) contains torsion, but such augmentations are *not* induced by any exact Lagrangian filling—even though their mod 2 reductions **are** geometric.

- For example: For \(\Lambda_k\) and \(n\) odd, \(n\neq \pm1\), the augmentation \(\epsilon_n\) is not geometric, though its reduction mod 2 is geometric.
- The Seidel isomorphism shows that if there *were* a geometric augmentation yielding such LCH with (more) torsion, the homology of the (fillings of genus 1) surface would not match.

### c. Role of Mod 2 Reductions

Even when an augmentation over \(\mathbb{Z}\) produces non-geometric behavior, its reduction mod 2 may still be geometric:

- For the above knots and augmentations, **the mod 2 reduction comes from an explicit decomposable filling** (constructed via saddle moves).
- However, being geometric mod 2 does **not** lift to being geometric over \(\mathbb{Z}\) in general, because of the emergence of torsion when reducing modulo divisors of \(n\).

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## 4. **Technical Insights and Broader Context**

### a. Augmentation Variety and Tangent Spaces

- The family of augmentations forms an algebraic variety; the Zariski tangent space at a point relates to LCH in degree 0 via the universal coefficient theorem.
- Torsion arises (for example) when a point in the augmentation variety over \(\mathbb{Z}/p\) is a singular point (intersection of components), leading to larger \(LCH^0\) and hence torsion by UCT.

### b. Positive Legendrian Knots

- For **positive** Legendrian knots (those isotopic to a presentation with only nonnegative Reeb chord gradings), LCH is always free and torsion-free for any augmentation (an explicit proposition and proof are provided).

### c. Uniqueness and Geography

- Connected sum constructions and the flexibility in augmentation parameters allow the realization of *any* finitely generated abelian group (in suitable degrees) as linearized contact homology.

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## 5. **Summary Table: Torsion Phenomena in Linearized LCH**

| Family | Augmentation | LCH over \(\mathbb{Z}\) | Lifting to filling? | Mod 2 geometric? |
|--------|--------------|-------------------------|---------------------|-----------------|
| \(\Lambda_k\) | \(n \neq 0, \pm1\) | Includes \(\mathbb{Z}/n\) | **No** (if \(n\) odd and \(\neq \pm1\)) | **Yes** |

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## 6. **Conclusions and Outlook**

- **Torsion in linearized LCH over \(\mathbb{Z}\) exists for Legendrian knots in \(\mathbb{R}^3\) and is robust: any abelian group can be realized as a summand (in certain degrees).**
- **Such torsion cannot arise from geometric (filling-induced) augmentations—augmentations supporting torsion are genuinely non-geometric over \(\mathbb{Z}\), though their reductions mod 2 can be geometric.**
- **This result distinguishes algebraic augmentations from those arising from embedded Lagrangian topology, and highlights the subtlety of lifting mod 2 behavior to the integers.**

The findings clarify the algebraic richness of Legendrian knot invariants and open lines of inquiry into the relationship between algebraic and geometric structures in contact topology.

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### *References to Definitions, Theorems, and Key Formulas in Text*
- **Proposition 1**: Statement that LCH with torsion summands exist for all \(n\).
- **General family construction**: Connected sum and clasping constructions for arbitrary torsion (see Figs 1–4 in the original).
- **Seidel isomorphism** (Proposition 3.4 of Gao–Rutherford): \[LCH_*^\epsilon(\Lambda;k) \cong H_{*+1}(L,\Lambda; k)\] for *geometric* augmentations.
- **Sabloff duality**: Relates \(LCH_i\) and \(LCH_{-i}\) (see Equations (2) and (3) in paper for precise formulations).

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**In summary:** 

Lipshitz and Ng demonstrate that there exist Legendrian knots in standard contact \(\mathbb{R}^3\) with augmentations over \(\mathbb{Z}\) such that their linearized contact homology contains torsion, and that such torsion cannot arise from exact Lagrangian fillings despite being consistent with geometric augmentations mod \(2\). This establishes a previously unobserved gap between algebraic and geometric aspects of Legendrian knot invariants.

Source: https://www.emergentmind.com/topics/linearized-legendrian-contact-homology-lch