Negative contact surgery on Legendrian non-simple knots

TL;DR

This paper demonstrates that negative rational contact surgeries on Legendrian non-simple knots can produce distinguishable contact 3-manifolds, even when classical invariants are identical, expanding understanding of Legendrian knot invariants.

Contribution

It provides the first examples of Legendrian knots with identical classical invariants but distinct contact surgeries for all negative rational numbers, generalizing to certain two-bridge knots.

Findings

Different contact 3-manifolds arise from surgeries on Legendrian knots with same classical invariants.

Negative rational contact surgeries with r ≠ -1 distinguish Legendrian knots via contact invariants.

Generalization to families of two-bridge knots broadens the scope of the results.

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 $r\neq -1$ 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.