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 develops two new invariants for primitive Legendrian knots in lens spaces L(p,q) with q ≠ 1. After reviewing the background of Legendrian knot theory—particularly the Chekanov–Eliashberg DGA for knots in ℝ³ and Sabloff’s DGA for knots in smooth S¹‑bundles over surfaces—the author focuses on the special topology of lens spaces, whose fundamental group is the finite cyclic group ℤ/pℤ. A “primitive” Legendrian knot is defined as one that does not represent a non‑trivial element of π₁(L(p,q)), a condition that guarantees a well‑behaved Lagrangian projection.

The first invariant is a differential graded algebra (DGA) constructed directly from a labeled Lagrangian projection of the pair (L(p,q), K). Each crossing in the projection receives an α‑ and a β‑label encoding both the tangent direction and the rotation number relative to the S¹‑bundle structure of the lens space. Using these labels, the grading (Maslov index) and the differential are defined exactly as in Sabloff’s construction: the differential counts holomorphic disks (or “Reeb chords”) whose boundaries run between crossings, weighted by signs determined from the labeling. The author proves that this DGA is invariant under Legendrian isotopy by showing that the labeling rules and disk counts are compatible with the Reidemeister moves adapted to the lens‑space setting. Consequently, the DGA provides a Legendrian isotopy invariant that captures information beyond classical invariants such as Thurston–Bennequin number and rotation number.

The second invariant exploits the p‑fold cyclic cover of the lens space, which is diffeomorphic to the three‑sphere S³. Lifting the primitive knot K to its preimage K̃ in S³ yields a Legendrian knot for which the standard Chekanov–Eliashberg DGA can be defined. The novelty lies in equipping this DGA with a free ℤ/pℤ‑action induced by deck transformations of the covering map. This action turns the DGA into a ℤ/pℤ‑equivariant object, encoding the symmetry of the original knot in the lens space. The author demonstrates that the equivariant DGA is a finer invariant: there exist pairs of primitive knots in L(p,q) that share the same non‑equivariant DGA but have distinct ℤ/pℤ‑actions, thus belonging to different Legendrian isotopy classes.

Computational techniques are presented in detail. For the first invariant, the paper outlines how to draw the labeled Lagrangian projection, assign Maslov gradings, and enumerate the relevant holomorphic disks. For the second invariant, the construction of the p‑fold cover, the explicit description of deck transformations, and the modification of the differential to respect the group action are all made algorithmic. Several concrete examples in L(5,2) illustrate the full process and confirm that the invariants distinguish knots that classical invariants cannot.

In the concluding section, the author emphasizes that these constructions open a new avenue for Legendrian knot theory in manifolds with finite fundamental groups. Potential extensions include the case q = 1 (where the lens space is a quotient of S³ by a cyclic group acting freely), higher‑dimensional lens spaces, and connections with Heegaard Floer homology or Khovanov‑type Legendrian invariants. The paper thus provides both a theoretical framework and practical tools for studying Legendrian knots beyond the traditional S³ setting.