Combinatorial Reeb dynamics on punctured contact 3–manifolds

Avdek, Russell¹

¹ Department of Mathematics, Uppsala University, Uppsala, Sweden

Geometry & topology, Tome 27 (2023) no. 3, pp. 953-1082.

Résumé

Let ${\Lambda }^{\pm }={\Lambda }^{+}\cup {\Lambda }^{-}\subset ({ℝ}^3,{\xi }_{\mathrm{std}})$ be a contact surgery diagram determining a closed, connected contact $3$–manifold $(S_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$ and an open contact manifold $({ℝ}_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$. Following work of Bourgeois, Ekholm and Eliashberg, we demonstrate how ${\Lambda }^{\pm }$ determines a family ${\alpha }_{𝜖}$ of contact forms for $({ℝ}_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$ whose closed Reeb orbits are in one-to-one correspondence with cyclic words of composable Reeb chords on ${\Lambda }^{\pm }$. 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 $({ℝ}_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$.

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, $\mathrm{tb}=1$ trefoil for $k>0$, which are known to have nonzero Heegaard Floer contact classes by work of Lisca and Stipsicz.

Détail

DOI : 10.2140/gt.2023.27.953

Keywords: contact surgery, contact homology, Legendrian knot

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