Combinatorial Reeb dynamics on punctured contact 3–manifolds

Avdek, Russell

doi:10.2140/gt.2023.27.953

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Combinatorial Reeb dynamics on punctured contact $3$–manifolds

Russell Avdek

Geometry & Topology 27 (2023) 953–1082

DOI: 10.2140/gt.2023.27.953

Abstract

Let $Λ^{\pm} = Λ^{+} \cup Λ^{-} \subset (ℝ^{3}, ξ_{std})$ be a contact surgery diagram determining a closed, connected contact $3$ –manifold $(S_{Λ^{\pm}}^{3}, ξ_{Λ^{\pm}})$ and an open contact manifold $(ℝ_{Λ^{\pm}}^{3}, ξ_{Λ^{\pm}})$ . Following work of Bourgeois, Ekholm and Eliashberg, we demonstrate how $Λ^{\pm}$ determines a family $α_{𝜖}$ of contact forms for $(ℝ_{Λ^{\pm}}^{3}, ξ_{Λ^{\pm}})$ whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on $Λ^{\pm}$ . We compute the homology classes and integral Conley–Zehnder indices of these orbits diagrammatically and develop algebraic tools for studying holomorphic curves in surgery cobordisms between the $(ℝ_{Λ^{\pm}}^{3}, ξ_{Λ^{\pm}})$ .

These new techniques are used to describe the first known examples of closed, tight contact manifolds with vanishing contact homology: they are contact $1 ∕ k$ surgeries along the right-handed, $tb = 1$ trefoil for $k > 0$ , which are known to have nonzero Heegaard Floer contact classes by work of Lisca and Stipsicz.

Keywords

contact surgery, contact homology, Legendrian knot

Mathematical Subject Classification

Primary: 53D42

Secondary: 57K33

References

Publication

Received: 8 July 2020

Revised: 11 October 2021

Accepted: 13 November 2021

Published: 8 June 2023

Proposed: András I Stipsicz

Seconded: Leonid Polterovich, Yakov Eliashberg

Authors

Russell Avdek
Department of Mathematics Uppsala University Uppsala Sweden
https://russellavdek.com

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Volume 27, issue 3 (2023)