Linearized Legendrian Contact Homology

• Linearized Legendrian Contact Homology (LCH) is defined via differential graded algebras (DGAs) and explicit augmentations that capture detailed torsion phenomena.
• By adjusting augmentation parameters, researchers can generate torsion of arbitrary order, distinguishing algebraic invariants from those induced by geometric Lagrangian fillings.
• Mod 2 reductions and the Seidel isomorphism provide key insights into the constraints on LCH ranks, highlighting the interplay between algebraic structures and geometric topology.

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.