On Contact Round Surgeries on $(\mathbb{S}^3,ξ_{st})$ and Their Diagrams

Abstract: We introduce the notions of contact round surgery of index 1 and 2, respectively, on Legendrian knots in $\left(\mathbb{S}^3, \xi_{st}\right)$ and associate diagrams to them. We realize Jiro Adachi's contact round surgeries as special cases. We show that every closed connected contact 3-manifold can be obtained by performing a sequence of contact round surgeries on some Legendrian link in $\left(\mathbb{S}^3, \xi_{st}\right)$, thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to a similar result of Ding and Geiges for contact Dehn surgeries. We discuss a bridge between certain pairs of contact round surgery diagrams of index 1 and 2 and contact $\pm1$-surgery diagrams. We use this bridge to establish the result mentioned above.