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Combinatorial Reeb dynamics on punctured contact $3$–manifolds

Russell Avdek

Geometry & Topology 27 (2023) 953–1082
Abstract

Let Λ± = Λ+ Λ (3,ξstd ) be a contact surgery diagram determining a closed, connected contact 3–manifold (SΛ±3,ξΛ±) and an open contact manifold (Λ±3,ξΛ±). Following work of Bourgeois, Ekholm and Eliashberg, we demonstrate how Λ± determines a family α𝜖 of contact forms for (Λ±3,ξΛ±) whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on Λ±. 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 (Λ±3,ξΛ±).

These new techniques are used to describe the first known examples of closed, tight contact manifolds with vanishing contact homology: they are contact 1k 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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