On Contact Round Surgeries on $(\mathbb{S}^3,ξ_{st})$ and Their Diagrams
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911338114056192 |
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| author | Deep, Prerak Kulkarni, Dheeraj |
| author_facet | Deep, Prerak Kulkarni, Dheeraj |
| contents | We introduce the notion of contact round surgery of index $1$ on Legendrian knots in a general contact 3-manifold. It generalizes the notion of contact round surgery of index 1 on Legendrian knots introduced by Adachi. In $\left(\mathbb{S}^3, ξ_{st}\right)$, we introduce the notion of contact round surgery of index 2 on a Legendrian knot and realize Adachi's contact round 2-surgery on a convex torus as a contact round surgery of index $2$ on a Legendrian knot in $\left(\s^3, ξ_{st}\right)$. We associate surgery diagrams to contact round surgeries of indices 1 and 2 on Legendrian knots in $\left(\mathbb{S}^3, ξ_{st}\right)$. With this set-up, 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 $\left(\mathbb{S}^3, ξ_{st}\right)$, thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to the result of Ding-Geiges for contact Dehn surgeries. We also discuss a bridge between certain pairs of contact round surgery diagrams of indices 1 and 2, and contact $(\pm1)$-surgery diagrams. We use this bridge to establish the result mentioned above. In the end, we derive a corollary that gives sufficient conditions on contact round surgeries to produce symplectically fillable manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Contact Round Surgeries on $(\mathbb{S}^3,ξ_{st})$ and Their Diagrams Deep, Prerak Kulkarni, Dheeraj Symplectic Geometry Geometric Topology 57K33, 57R65 We introduce the notion of contact round surgery of index $1$ on Legendrian knots in a general contact 3-manifold. It generalizes the notion of contact round surgery of index 1 on Legendrian knots introduced by Adachi. In $\left(\mathbb{S}^3, ξ_{st}\right)$, we introduce the notion of contact round surgery of index 2 on a Legendrian knot and realize Adachi's contact round 2-surgery on a convex torus as a contact round surgery of index $2$ on a Legendrian knot in $\left(\s^3, ξ_{st}\right)$. We associate surgery diagrams to contact round surgeries of indices 1 and 2 on Legendrian knots in $\left(\mathbb{S}^3, ξ_{st}\right)$. With this set-up, 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 $\left(\mathbb{S}^3, ξ_{st}\right)$, thus obtaining a contact round surgery diagram for each contact 3-manifold. This is analogous to the result of Ding-Geiges for contact Dehn surgeries. We also discuss a bridge between certain pairs of contact round surgery diagrams of indices 1 and 2, and contact $(\pm1)$-surgery diagrams. We use this bridge to establish the result mentioned above. In the end, we derive a corollary that gives sufficient conditions on contact round surgeries to produce symplectically fillable manifolds. |
| title | On Contact Round Surgeries on $(\mathbb{S}^3,ξ_{st})$ and Their Diagrams |
| topic | Symplectic Geometry Geometric Topology 57K33, 57R65 |
| url | https://arxiv.org/abs/2504.06074 |