Invariants for Legendrian knots in lens spaces

In this paper we define invariants for primitive Legendrian knots in lens spaces L(p,q) for q not equal to 1. The main invariant is a differential graded algebra which is computed from a labeled Lagrangian projection of the pair (L(p,q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S^1-bundles over Riemann surfaces. The second invariant defined for knots in lens spaces takes the form of a DGA enhanced with a free cyclic group action and can be computed from the p-fold cover of the pair (L(p,q), K).

💡 Research Summary

The paper introduces two new differential graded algebra (DGA) invariants for primitive Legendrian knots in lens spaces L(p,q) with q≠1. The motivation stems from the fact that existing combinatorial invariants—Chekanov–Eliashberg’s DGA for Legendrian knots in standard contact ℝ³ and Sabloff’s DGA for Legendrian knots in smooth S¹‑bundles over Riemann surfaces—do not apply directly to the non‑trivial lens spaces L(p,q) when q≠1. The authors overcome this limitation by exploiting the covering relationship between a primitive knot K⊂L(p,q) and its freely p‑periodic lift ẽK⊂S³, which is a p‑fold cover under the standard cyclic action defining the lens space.

The first invariant is constructed directly from a labeled Lagrangian projection Γ of the pair (L(p,q),K) onto the 2‑sphere. Each crossing of Γ is assigned two generators a_i and b_i, together with a choice of preferred Reeb chord and a sign convention that records which quadrant of the crossing is filled. The authors define a “defect” for each region of the diagram: defect = (1/2π)∫_region Ω + Σ ε(x)·ℓ(x), where Ω is the curvature form on S², ε(x) = ±1 depending on the sign of the generator, and ℓ(x) is the length of the corresponding Reeb chord (normalized so that a full fiber has length 1). This defect measures the winding of the lifted curve around the Reeb fiber and ensures that the projection encodes the full Legendrian isotopy class.

Using the generators and the defect, they build the tensor algebra A = T(a₁,b₁,…,a_n,b_n) and define a differential ∂ by counting immersed “admissible disks” in S² whose boundary follows the diagram and whose defect vanishes. The differential satisfies ∂²=0 and lowers the cyclic grading by one. The resulting pair (A,∂) is a semi‑free DGA graded by ℤ_p, and the authors prove (Theorem 2) that its stable tame isomorphism class is invariant under Legendrian isotopy of K.

The second invariant leverages Sabloff’s low‑energy DGA (ẽA,ẽ∂) for the lift ẽK⊂S³. Because every Reeb orbit in the standard contact S³ fibers over S², the low‑energy algebra is finitely generated by chords shorter than a full fiber. The authors endow (ẽA,ẽ∂) with a free cyclic group action γ of order p that cyclically permutes the generators corresponding to the p‑fold symmetry of the lift. They then define an equivariant DGA (ẽA,γ,ẽ∂) and introduce a notion of Z_p‑equivariant stable tame equivalence (including Z_p‑stabilizations and Z_p‑tame isomorphisms). Theorem 3 states that the equivariant DGA class is an invariant of the Legendrian type of K.

A substantial technical portion of the paper is devoted to showing that (A,∂) can be identified with the Z_p‑invariant sub‑algebra of (ẽA,ẽ∂) and that the differential structures are compatible with the group action. This involves careful analysis of the defect formula, the grading conventions, and the behavior of admissible disks under the covering map. The authors also construct explicit Z_p‑stabilizations by adding pairs of generators (e₁,i, e₂,i) that transform under γ, ensuring that the equivariant structure is preserved after stabilization.

The final section presents explicit computations for knots in L(3,2) and L(5,2). For each example, the authors draw the labeled Lagrangian projection, compute the defects, list the generators, and write down the differential. They verify that the resulting DGAs are non‑trivial and distinguish knots that have identical classical invariants (Thurston–Bennequin number and rotation number). These examples illustrate the practical computability of both invariants and demonstrate that the equivariant DGA captures information beyond the classical Legendrian invariants.

In summary, the paper provides two robust algebraic tools for studying Legendrian knots in non‑trivial lens spaces: a direct diagrammatic semi‑free DGA and an equivariant DGA obtained from the p‑fold cover in S³. Both invariants are shown to be well‑defined under Legendrian isotopy, extend the reach of Chekanov–Eliashberg and Sabloff’s constructions, and open the way for further applications of DGA techniques to contact 3‑manifolds with more complicated topology.