Stein traces and characterizing slopes

Abstract

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

Contact surgery numbers

“…Again, a contact handle slide can be seen as an isotopy of K in the contact manifold obtained by surgery along K 1 and thus the same proof works for other coefficients of K as well. In this setting the contact surgery coefficient stays the same, see for example [CEK21]. So we also obtain the other inequalities.…”

“…If we move a small part of K near K 1 and slide it over the newly glued-in solid torus the knot K will deform to the knot K 2 exactly as in Figure 9, cf. the proof of the contact handle slide in [CEK21]. The above slight adoption of the proof from [GY10] generalizes to the setting of contact manifolds by using the results from Section 2.…”

“…We will also need the following version of a handle slide for contact surgery which is due to Ding and Geiges [DG09], cf. [CEK21] for the generalization to other contact surgery coefficients.…”

Contact Surgery Graphs

“…We assume all 3-manifolds to be connected, closed, oriented and smooth; all contact structures are positive and coorientable. For background on contact surgery and symplectic and Stein cobordisms, we refer to [2,4,6,8,16,17,20,22,32]. Legendrian links in (S 3 , ξ st ) are always presented in their front projection.…”

Contact surgery graphs

“…Legendrian links in (S 3 , ξ st ) are always presented in their front projection. We choose the normalization of the d 3 -invariant as in [CEK21,EKO22] which differs from the normalizations in [Go98, DGS04, DK16] by 1/2. Using our normalization we see that contact structures on homology spheres have integral d 3 -invariants (in particular d 3 (S 3 , ξ st ) = 0) and that the d 3 -invariant is additive under connected sums.…”