The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained-they correspond to open handlebodies with all handles of index < 2. An uncountable collection of exotic l4,'S is shown to admit Stein structures. New invariants of contact 3-manifolds are produced, including a complete (and computable) set of invariants for determining the homotopy class of a 2-plane field on a 3-manifold. These invariants are applicable to Seiberg-Witten theory. Several families of oriented 3-manifolds are examined, namely the Seifert fibered spaces and all surgeries on various links in S3, and in each case it is seen that "most" members of the family are the

Handlebody construction of Stein surfaces

We develop new techniques in the theory of convex surfaces to prove complete classification results for tight contact structures on lens spaces, solid tori, and T2×I.

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On the classification of tight contact structures I

Let the space R = {(q, p, u)} be equipped with the standard contact form α = du − pdq. A link L ⊂ R3 is called Legendrian if the restriction of α to L vanishes. Two Legendrian links are said to be Legendrian isotopic if they can be connected by a path in the space of Legendrian embeddings. Consider the map π : R3 → R2, (q, p, u) → (q, p). Since the kernel of dπ is everywhere transverse to the kernel of α, the projection π(L) of a Legendrian link L is an immersed curve. We say that a link L ⊂ R3 is generic with respect toπ, or π-generic, if all self-intersections of π(L) are transverse double points. Given a π-generic Legendrian link L , its diagram is the curve π(L) ⊂ R2, at every crossing of which the overpassing branch (the one with the larger value of u) is marked. It is well known that every smooth link type can be realized by a Legendrian link (it is shown in Sect. 11 that such a realization can be chosen to have a diagram with relatively few crossings). The so-called classical invariants of an oriented Legendrian link L are defined as follows. First of them is just the smooth isotopy type of L . Let L1, . . . , Lk be the components of L . The Maslov number m(Li) of Li is twice the rotation number of π(Li). The Thurston–Bennequin number β(Li) of Li is the linking number between Li and s(Li), where s is a small shift along the u direction. The orientation on R3 involved in the definition of the linking number is given by the form α ∧ dα = dq ∧ dp ∧ du. In terms of the diagram, β(Li) is the number of double points of π(Li) counted with signs shown in Fig. 1. The Thurston–Bennequin number does not depend on the orientation of Li , while the Maslov number changes its sign when the orientation changes.

Differential algebra of Legendrian links

Publisher Summary Contact structures on manifolds and Legendrian and transversal knots in them are very natural objects, born over a century ago, in the work of Huygens, Hamilton, and Jacobi on geometric optics and work of Lie on partial differential equations. This chapter discusses Legendrian and transversal knots in dimension three where their theory is most fully developed and where they are most intimately tied to topology. In this dimension a predominately topological and combinatorial approach may be used to their study. The chapter then examines higher dimensional Legendrian knots. The chapter also explains the concept of Lagrangian protection. Analogous to the definition of finite type invariants of topological knots, finite type invariants of Legendrian knots can be defined. It is then possible to get a global contact isotopy if there is a fixed over twisted disk in the complement of both Legendrian knot. A complete understanding of Legendrian unknots in over twisted contact structures must wait for future work.

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Legendrian and Transversal Knots

We classify Legendrian torus knots and Legendrian figure eight knots in the tight contact structure on S3 up to Legendrian isotopy. A a corollary to this we also obtain the classification of transversal torus knots and transversal figure eight knots up to transversal isotopy.

https://www.intlpress.com/site/pub/files/_fulltext/journals/jsg/2001/0001/0001/JSG-2001-0001-0001-a003.pdf

Knots and Contact Geometry I: Torus Knots and the Figure Eight Knot

Classification of topologically trivial Legendrian knots

The first part of this paper contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e., Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. These techniques were never published in detail when the classification result was announced over ten years ago. The final part of the present paper contains a systematic discussion of Legendrian knots in overtwisted contact manifolds, and in particular, gives the coarse classification (i.e., classification up to a global contactomorphism) of topologically trivial exceptional Legendrian knots in overtwisted contact $S^3$ according to the values of the invariants tb, r. We show, moreover, that such knots only occur for one of the infinitely many overtwisted contact structures on $S^3$. We remark that our tight classification result also implies that any topologically trivial loose Legendrian knots with same value of (tb, r) in an overtwisted contact 3-manifold are in fact Legendrian isotopic if $tb < 0$.

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Topologically trivial Legendrian knots

The human cortex exhibits intrinsic neural timescales that shape a temporal hierarchy. Whether this temporal hierarchy follows the spatial hierarchy of its topography, namely the core-periphery organization, remains an open issue. Using magnetoencephalography data, we investigate intrinsic neural timescales during rest and task states; we measure the autocorrelation window in short (ACW-50) and, introducing a novel variant, long (ACW-0) windows. We demonstrate longer ACW-50 and ACW-0 in networks located at the core compared to those at the periphery with rest and task states showing a high ACW correlation. Calculating rest-task differences, i.e., subtracting the shared core-periphery organization, reveals task-specific ACW changes in distinct networks. Finally, employing kernel density estimation, machine learning, and simulation, we demonstrate that ACW-0 exhibits better prediction in classifying a region’s time window as core or periphery. Overall, our findings provide fundamental insight into how the human cortex’s temporal hierarchy converges with its spatial core-periphery hierarchy. Golesorkhi et al. use a combination of magnetoencephalography data, machine learning and simulation to investigate intrinsic neural timescales during resting and task states. They provide insight into how the temporal hierarchy of the human cortex converges with its spatial core-periphery hierarchy.

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Temporal hierarchy of intrinsic neural timescales converges with spatial core-periphery organization.

For certain contact manifolds admitting a 1-periodic Reeb flow we construct a conjugation-invariant norm on the universal cover of the contactomorphism group. With respect to this norm the group admits a quasi-isometric monomorphism of the real line. The construction involves the partial order on contactomorphisms and symplectic intersections. This norm descends to a conjugation-invariant norm on the contactomorphism group. As a counterpoint, we discuss conditions under which conjugation-invariant norms for contactomorphisms are necessarily bounded.

Maia Fraser

Papers

Classification of topologically trivial Legendrian knots

Topologically trivial Legendrian knots

Temporal hierarchy of intrinsic neural timescales converges with spatial core-periphery organization.

On Sandon-type metrics for contactomorphism groups

On Sandon-type metrics for contactomorphism groups