Combinatorial Reeb dynamics on punctured contact 3-manifolds

Abstract

Let ${\Lambda }^{\pm }={\Lambda }^{+}\cup {\Lambda }^{-}\subset ({\mathbb{R}}^3,{\xi }_{std})$ be a contact surgery diagram determining a closed, connected contact $3$-manifold $(S_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$ and an open contact manifold $({\mathbb{R}}_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$. Following arXiv:0911.0026 and arXiv:1906.07228 we demonstrate how ${\Lambda }^{\pm }$ determines a family ${\alpha }_{\epsilon }$ of contact forms on $({\mathbb{R}}_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$ 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 $({\mathbb{R}}_{{\Lambda }^{\pm }}^3,{\xi }_{{\Lambda }^{\pm }})$. These new techniques are used to describe the first known examples of closed, tight contact manifolds with vanishing contact homology. They are contact $\frac{1}{k}$ surgeries along the right-handed, $tb=1$ trefoil for $k>0$, which are known to have non-zero Heegaard-Floer contact classes by arXiv:math/0404135.

91 pages, 32 figures. V5: Minor updates

References in corpus (5)

• Contact homology and virtual fundamental cycles
• Lectures on Symplectic Field Theory
• Semi-global Kuranishi charts and the definition of contact homology
• Holomorphic curves for Legendrian surgery
• Cobordism maps in embedded contact homology

Cited by in corpus (2)

• Exact Lagrangians in four-dimensional symplectisations
• Simplified SFT moduli spaces for Legendrian links