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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., Comment: 89 pages, 5 figures. This paper continues the work initiated in arXiv:2306.02125 to encompass all positive torus knots and provide topological and dynamical applications. In v2, we expanded our discussion of related work and made edits to the exposition

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., Comment: 77 pages, arXiv insists the primary is GT rather than SG, v2 revised and streamlined per referee's suggestions

We compute the cylindrical contact homology of the links of the simple singularities. These manifolds are contactomorphic to $S^3/G$ for finite subgroups $G\subset SU(2)$. We perturb the degenerate contact form on $S^3/G$ with a Morse function, which is invariant under the corresponding $H\subset SO(3)$ action on $S^2$, to achieve nondegeneracy up to an action threshold. The cylindrical contact homology is recovered by taking a direct limit of the action filtered homology groups. The ranks of this homology are given in terms of $|\text{Conj}(G)|$, demonstrating a Floer theoretic McKay correspondence., Comment: 52 pages, 9 figures, revised as suggested by referee

We obtain new sharp obstructions to symplectic embeddings of four-dimensional polydisks $P(a,1)$ into four-dimensional ellipsoids $E(bc,c)$ when $1\le a< 2$ and $b$ is a half-integer. When $1 \leq a < 2-O(b^{-1})$ we demonstrate that $P(a,1)$ symplectically embeds into $E(bc,c)$ if and only if $a+b\le bc$. Our results show that inclusion is optimal and extend the result by Hutchings \cite{H} when $b$ is an integer. Our proof is based on a combinatorial criterion developed by Hutchings \cite{H} to obstruct symplectic embeddings., Comment: 38 pages, to appear in Journal of Fixed Point Theory and Applications. Additional details are provided on the subtleties of the various convex generators used in the Hutchings criterion to obtain obstructions. More details on the suboptimal nature of obstructions coming from ECH capacities are also given in S 1.2

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse-Bott direct limit argument and the classification of certain J-holomorphic buildings. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings-Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings-Nelson., Comment: 87 pages, to appear in the Journal of Symplectic Geometry

In a previous paper, we showed that the original definition of cylindrical contact homology, with rational coefficients, is valid on a closed three-manifold with a dynamically convex contact form. However we did not show that this cylindrical contact homology is an invariant of the contact structure. In the present paper, we define "nonequivariant contact homology" and "S^1-equivariant contact homology", both with integer coefficients, for a contact form on a closed manifold in any dimension with no contractible Reeb orbits. We prove that these contact homologies depend only on the contact structure. Our construction uses Morse-Bott theory and is related to the positive S^1-equivariant symplectic homology of Bourgeois-Oancea. However, instead of working with Hamiltonian Floer homology, we work directly in contact geometry, using families of almost complex structures. When cylindrical contact homology can also be defined, it agrees with the tensor product of the S^1-equivariant contact homology with Q. We also present examples showing that the S^1-equivariant contact homology contains interesting torsion information. In a subsequent paper we will use obstruction bundle gluing to extend the above story to closed three-manifolds with dynamically convex contact forms, which in particular will prove that their cylindrical contact homology has a lift to integer coefficients which depends only on the contact structure., Comment: 93 pages; v3 corrected a typo and added a reference to the introduction; to appear in Journal of Topology

In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical set. This requires a mix of analytic arguments (establishing properties of the moduli spaces and evaluation maps) and formal arguments (defining or computing invariants from the analytic data). The goal of this paper is to isolate the formal arguments, in the case when the critical set is a union of circles. Namely, we state axioms for moduli spaces and evaluation maps (encoding a minimal amount of analytical information that one needs to verify in any given Floer-theoretic situation), and using these axioms we define homological invariants. More precisely, we define a (almost) category of "Morse-Bott systems". We construct a "cascade homology" functor on this category, based on ideas of Bourgeois and Frauenfelder, which is "homotopy invariant". This machinery is used in our work on cylindrical contact homology., Comment: 48 pages (v3 has minor clarifications, mainly at the end, following referee's suggestions)

This paper is the sequel to the previous paper [Ne15], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder indices with respect to a free homotopy class; see Definition 1.7. {Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits.} We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, {and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure.} We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [El07] in dimension three., Comment: 76 pages, incorporates changes helpfully suggested by the referee. More details on filtered cylindrical contact homology and the embedded contact homology apparatus used to prove the main regularity results are given

This is a light survey article about the origins of contact and symplectic topology in dynamics and the more recent developments in the field. In lieu of formulas, numerous anecdotes are given., Comment: 10 pages. Many thanks are due to Alan Weinstein and Helmut Hofer for sharing their recollections on early symplectic history

In this paper we obtain new obstructions to symplectic embeddings of the four-dimensional polydisk $P(a,1)$ into the ball $B(c)$ for $2\leq a<\frac{\sqrt{7}-1} {\sqrt{7}-2} \approx 2.549$, extending work done by Hind-Lisi and Hutchings. Schlenk's folding construction permits us to conclude our bound on $c$ is optimal. Our proof makes use of the combinatorial criterion necessary for one "convex toric domain" to symplectically embed into another introduced by Hutchings in \cite{Beyond}. Additionally, we prove that if certain symplectic embeddings of four dimensional convex toric domains exist then a modified version of this criterion from \cite{Beyond} must hold, thereby reducing the computational complexity of the original criterion from $O(2^n)$ to $O(n^2)$., Comment: 31 pages, revised as suggested by referee and M. Hutchings

The link of the $A_n$ singularity, $L_{A_n} \subset \mathbb{C}^3$ admits a natural contact structure $\xi_0$ coming from the set of complex tangencies. The canonical contact form $\alpha_0$ associated to $\xi_0$ is degenerate and thus has no isolated Reeb orbits. We show that there is a nondegenerate contact form for a contact structure equivalent to $\xi_0$ that has two isolated simple periodic Reeb orbits. We compute the Conley-Zehnder index of these simple orbits and their iterates. From these calculations we compute the positive $S^1$-equivariant symplectic homology groups for $\left(L_{A_n}, \xi_0 \right)$. In addition, we prove that $\left(L_{A_n}, \xi_0 \right)$ is contactomorphic to the Lens space $L(n+1,n)$, equipped with its canonical contact structure $\xi_{std}$., Comment: 23 pages, improvements to exposition, this is the final version to appear in Involve

This paper helps to clarify the status of cylindrical contact homology, a conjectured contact invariant introduced by Eliashberg, Givental, and Hofer in 2000. We explain how heuristic arguments fail to yield a well-defined homological invariant in the presence of multiply covered curves. We then introduce a large subclass of dynamically convex contact forms in dimension 3, termed dynamically separated, and demonstrate automatic transversality holds, therby allowing us to define the desired chain complex. The Reeb orbits of dynamically separated contact forms satisfy a uniform growth condition on their Conley-Zehnder index under iteration, typically up to large action; see Definition 1.15 These contact forms arise naturally as perturbations of Morse-Bott contact forms such as those associated to $S^1$-bundles. In subsequent work, we give a direct proof of invariance for this subclass and, when further proportionality holds between the index and action, powerful geometric computations in a wide variety of examples., Comment: 68 pages, added more information about bad Reeb orbits, added a proof of a beloved folk theorem concerning the factorization of multiply covered curves, contains expository revisions helpfully suggested by the referee

We show that for dynamically convex contact forms in three dimensions, the cylindrical contact homology differential d can be defined by directly counting holomorphic cylinders for a generic almost complex structure, without any abstract perturbation of the Cauchy-Riemann equation. We also prove that d^2 = 0. Invariance of cylindrical contact homology in this case can be proved using S^1-dependent almost complex structures, similarly to work of Bourgeois-Oancea; this will be explained in another paper., Comment: v3: corrected Lemma 2.5(b), to appear in Journal of Symplectic Geometry

This report summarizes three related studies of 3- to 5-year-old children's temperament and its relationship to their social competence, ability to solve interpersonal problems, locus of control, parent behavior and teacher/child interactions. Fifty-eight children, predominantly middle class participants in a laboratory preschool, and their parents were subjects for portions of this study. A trained observer used a "focal person" method to record interactions between the head teacher and individual children in two groups of 19 children. The 24 four-year-olds in this study were also rated by parents and teachers on the Thomas, Chess and Korn Temperament Scale, and were assessed according to the Stanford Preschool Internal-External Scale and the Preschool Interpersonal Problem Solving Scale. Another group of 20 children were subjects of a correlational study of parents' self-ratings on the Iowa Parent Behavior Inventory and ratings of their children on the temperament questionnaire and the Iowa Social Competency Scale. Results suggested differences between mothers and fathers in the ways they see and interpret their children and indicated that individual differences in children's behavorial style were related to different patterns of teacher/child interaction. There were also correlations between temperament variables and the children's sex, age, and birth order. Results also indicated either that the temperament questionnaire is unreliable across settings or that temperament is setting-specific. (CB)

This study examined the relationship between level of ego development and parents' self reports of the subjective experience of parenthood. Subjects were 30 mother/father pairs of middle class parents, 10 from each of three phases of the family: families whose oldest children were attending preschool or middle school, or had graduated from high school. Two assessment procedures were used: (1) an individual interview assessed parents' perceptions of the experience of parenthood, and (2) Loevinger's Washington University Sentence Completion Test Form 11-68 for men and women assessed level of ego development. Specifically investigated was the relationship of ego devlopment to response categories of self awareness, valuing interpersonal relationships, individuality, affect, empathy, and insight. With the exception of themes of individuality and empathy, the above categories of response were significantly related to parents' ego development. Other response themes also were identified. Ego level did not appear to increase with family phase. Results demonstrated that mothers and fathers varying in ego development describe the subjective experience of parenthood in qualitatively different ways. It is tentatively concluded, along the lines of Loevinger's model of ego development, that parents' responses to childrearing contain many elements that are a touchstone of ego development. (RH)