Distinctness of contact manifolds from Legendrian surgery on stabilized Chekanov–Eliashberg twist knots

Determine whether contact (−1)-surgery (Legendrian surgery) on stabilized Legendrian representatives of the Chekanov–Eliashberg twist knots E_n in the standard tight contact structure on S^3 produces different contact 3-manifolds up to contact isotopy. Equivalently, ascertain whether negative integer contact surgeries on non-stabilized Legendrian representatives of E_n yield distinct contact 3-manifolds.

Background

Etnyre posed the general question of whether Legendrian surgery always produces distinct contact manifolds, and specifically for the Chekanov–Eliashberg twist knots E_n. Subsequent work by Bourgeois–Ekholm–Eliashberg established distinctness for Legendrian surgery on certain non-stabilized, max–tb representatives of E_n, leveraging linearized contact homology.

However, the stabilized setting is fundamentally different: the Legendrian DGA vanishes for stabilized Legendrians, preventing a direct application of the Bourgeois–Ekholm–Eliashberg argument. The paper addresses many negative rational surgeries (r ≠ −1) via Heegaard Floer contact invariants and LOSS invariants, but the case of Legendrian surgery (r = −1) on stabilized twist knots remains unresolved in general.