Stein traces and characterizing slopes

We show that there exists an infinite family of pairwise non-isotopic Legendrian knots in the standard contact 3-sphere whose Stein traces are equivalent. This is first example such phenomenon. Different constructions developed article, including a annulus twist, explicit Weinstein handlebody equivalences, and discussion on dualizable patterns setting. These can be used to systematically construct distinct with contactomorphic $$(-1)$$ -surgeries and, many cases, equivalent traces. In addition, we also discuss characterizing slopes provide results opposite direction, i.e. describe cases which trace, or contactomorphism type r-surgery, uniquely determines isotopy type.