Torus knotted Reeb dynamics and the Calabi invariant

TL;DR

This work connects three-dimensional contact topology with two-dimensional surface dynamics by computing knot-filtered embedded contact homology for the positive torus knot T(p,q) in S^3. Using a Morse–Bott framework and careful ECH index calculations, the authors derive explicit action and linking bounds for Reeb orbits around a knotted elliptic core, and obtain obstructions to exact symplectic cobordisms between torus knots. They translate these Reeb-dynamics results into quantitative bounds for area-preserving diffeomorphisms of genus g = (p−1)(q−1)/2 surfaces with one boundary, via the Calabi invariant. The approach relies on non-toric, Morse–Bott perturbations and a detailed knot filtration on ECH, yielding both new invariants (c_k, c_k^{link}) and concrete dynamical consequences with potential broad applicability to open-book–supported contact structures.

Abstract

We establish the existence of a secondary Reeb orbit set with quantitative action and linking bounds for any contact form on the standard tight three-sphere admitting the standard transverse positive $T(p,q)$ torus knot as an elliptic Reeb orbit with a canonically determined rotation number. This can be interpreted through an ergodic lens for Reeb flows transverse to a surface of section. Our results also allow us to deduce an upper bound on the mean action of periodic orbits of naturally associated classes of area preserving diffeomorphisms of the associated Seifert surfaces of genus $(p-1)(q-1)/2$ in terms of the Calabi invariant, without the need for genericity or Hamiltonian hypotheses. Our proofs utilize knot filtered embedded contact homology, which was first introduced and computed by Hutchings for the standard transverse unknot in the irrational ellipsoids and further developed in our previous work. We continue our development of nontoric methods for embedded contact homology and establish the knot filtration on the embedded contact homology chain complex of the standard tight three-sphere with respect to positive $T(p,q)$ torus knots, where there are nonvanishing differentials. We also obtain obstructions to the existence of exact symplectic cobordisms between positive transverse torus knots.

Figures (5)

• Figure 1.1: Here we have depicted the gradient flow of $\mathpzc{H}_{p,q}$ on ${\mathbb C}{\mathbb P}^1_{p,q}$ for $p\neq2$ and labeled the fibers of the prequantization orbibundle projecting to the respective critical points. The minima have isotropy ${\mathbb Z}/p$ and isotropy ${\mathbb Z}/q$; the fibers $\mathpzc{p}$ and $\textcolor{bg}{\mathpzc{q}}$ project to the respective minima. The fiber $\mathpzc{h}$ projects to the index 1 critical point. The binding fiber $\mathpzc{b}$ projects to the maximum.
• Figure 3.1: Here we have depicted the gradient flow of $\mathpzc{H}_{\mathpzc{p},\mathpzc{q}}$ on ${\mathbb C}{\mathbb P}^1_{p,q}$ and labelled the fibers of the prequantization orbibundle projecting to the critical points. The minima have isotropy ${\mathbb Z}/p$ and isotropy ${\mathbb Z}/q$; the fibers $\mathpzc{p}$ and $\textcolor{bg}{\mathpzc{q}}$ project to the respective minima. The fiber $\mathpzc{h}$ projects to the index 1 critical point. The binding fiber $\mathpzc{b}$ projects to the maximum. The arrangement of these gradient flow lines is key to the proof of Proposition \ref{['prop:pqM']}.
• Figure 5.1: The path $\Lambda^-(q)$ for $(p,q)=(3,4)$ is in solid dark blue; it connects the origin to $(q,\lceil pq-q\delta_{\mathpzc{q},L}\rceil)=(q,pq)$. The line $y=(3/4-\delta_{\mathpzc{q},L})x$ is in dashed light blue. Because $p$ and $q$ are coprime, we have that for the embedded Reeb orbit $\mathpzc{q}$, $P^-_{p/q-\delta_{\mathpzc{q},L}}(q)=(q)$.
• Figure 5.2: On the left in red, blue, and green we have drawn the curve $\Lambda$ representing the generator $\mathpzc{h}\mathpzc{p}\mathpzc{q}$ when $(p,q)=(3,4)$. On the right in red and green is the curve $\Lambda_\mathpzc{q}$ obtained by rounding the pink corner at $(1,4)$, which represents $\mathpzc{p}\mathpzc{q}^5$ and appears in $\partial\Lambda$. (The reader may compute their ECH indices by counting lattice points, and compare the result with Table \ref{['table:genp']} (a).) In both figures the dashed light blue line has slope $-3/4$.
• Figure 5.3: This figure shows the (dotted light blue) line determining the action and knot filtration value of the generators $\mathpzc{p}\mathpzc{q}^5, \mathpzc{p}^4\mathpzc{q}, \mathpzc{h}\mathpzc{p}\mathpzc{q}$, and $\mathpzc{b}\mathpzc{p}\mathpzc{q}$ corresponding to the curves illustrated in (a)-(c), respectively. (Note that the curve in (c) represents both $\mathpzc{b}\mathpzc{p}\mathpzc{q}$ and $\mathpzc{h}\mathpzc{p}\mathpzc{q}$ as its middle blue segment is unlabeled.)

Theorems & Definitions (154)

• Theorem 1.1: Taubes
• Remark 1.2: Comparison with knot embedded contact homology
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Corollary 1.7
• Theorem 1.8
• Remark 1.9
• Remark 1.10: Comparison to combinatorial toric methods