Negative contact surgery on Legendrian non-simple knots

Abstract

We prove that for any pair of Legendrian representatives of the Chekanov-Eliashberg twist knots with different LOSS invariants, any negative rational contact $r$-surgery with $r\ne -1$ always gives rise to different contact 3-manifolds distinguished by their contact invariants. This gives the first examples of pairs of Legendrian knots with the same classical invariants but distinct contact $r$-surgeries for all negative rational number $r$. We also generalize the statement from the twist knots to a certain families of two-bridge knots.

v2: revised according to the comments of the referee; this is the final version to appear in Journal of Symplectic Geometry