Torus knot filtered embedded contact homology of the tight contact 3-sphere

Abstract

Knot filtered embedded contact homology was first introduced by Hutchings in 2015; it has been computed for the standard transverse unknot in irrational ellipsoids by Hutchings and for the Hopf link in lens spaces L(n,n-1) via a quotient by Weiler. While toric constructions can be used to understand the ECH chain complexes of many contact forms adapted to open books with binding the unknot and Hopf link, they do not readily adapt to general torus knots and links. In this paper, we generalize the definition and invariance of knot filtered embedded contact homology to allow for degenerate knots with rational rotation numbers. We then develop new methods for understanding the embedded contact homology chain complex of positive torus knotted fibrations of the standard tight contact 3-sphere in terms of their presentation as open books and as Seifert fibered spaces. We provide Morse-Bott methods, using a doubly filtered complex and the energy filtered perturbed Seiberg-Witten theory developed by Hutchings and Taubes, and use them to compute the T(2,q) knot filtered embedded contact homology, for q odd and positive. In the sequel we complete the computation for positive T(p,q) knots (where there is a nonvanishing differential) and use our results to deduce quantitative existence results for torus knotted Reeb dynamics on the tight 3-sphere and the mean action of area preserving diffeomorphisms of once punctured surfaces of arbitrary genus arising as Seifert surfaces of positive torus knots.

Figures (4)

• Figure 2.1: (The construction and gradient flow of $H_{2,3}$ on ${\mathbb C}{\mathbb P}^1_{2,3}$.) Left and Center: On the left is the fundamental domain for the punctured torus, where edges are identified according to their color (with no twist). The wedge is the fundamental domain for the action of the Reeb return map, which acts on by a clockwise $\frac{2\pi}{6}$ rotation. Right: We depict ${\mathbb C}{\mathbb P}^1_{2,3}$ obtained by gluing the center picture and closing up the puncture by collapsing it to a maximum (depicted by a black dot). Gluing in the associated solid torus produces the binding fiber $b$ of the open book. Linking of fibers: When glued as indicated on the left, the gray dot appears $2=\ell(b,e)$ times while the pink dot appears $3=\ell(b,h)$ times.
• Figure : (Injectivity) Let $\Psi(\sigma)=0$. Then there exists $L_0$ such that $\widetilde{\Psi(\sigma)} =0$ for some $\varepsilon_0$. We can map $\widetilde{\Psi(\sigma)}$ to zero under $\Phi^{L_0}$ from \ref{['eq:maps']}, thus $\sigma \sim 0$.
• Figure : (Injectivity) Let $\Psi(\sigma)=0$. Then there exists $L_0$ such that $\widetilde{\Psi(\sigma)} =0$ for some $\varepsilon_0$. We can map $\widetilde{\Psi(\sigma)}$ to zero under $\Phi^{L_0}$ from \ref{['eq:maps']}, thus $\sigma \sim 0$.
• Figure : (Surjectivity) Let $\sigma_0$ be some representative of an element $\tau$ for some $\varepsilon_0$. By taking the limit as $\varepsilon \to 0$, the image of $\sigma_0$ is $\sigma'$. Then $[\tau] = \Psi([\sigma'])$.

Theorems & Definitions (140)

• Definition 1.1
• Remark 1.2: The role of degree
• Theorem 1.3: Taubes
• Remark 1.4: Comparison with knot embedded contact homology
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8: Threshold between grading and filtration with $\delta$
• Remark 1.9: Generalization to $T(p,q)$ for $p\neq 2$
• Remark 1.10: Comparison to a toric perturbation