Lisa Traynor

Generating function polynomials for legendrian links

It is shown that, in the 1{jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1{jet of the 0{function, and thus cannot be distinguished by the classical rotation number or Thurston{Bennequin invariants. The links are distinguished by calculating invariant polynomials dened via homology groups associated to the links through the theory of generating functions. The many calculations of these generating function polynomials support the belief that these polynomials carry the same information as a rened version of Chekanov’s rst order polynomials which are dened via the theory of holomorphic curves.

Lagrangian cobordisms via generating families: Construction and geography

Embedded Lagrangian cobordisms between Legendrian submanifolds are produced by isotopy, spinning, and handle-attachment constructions that employ the technique of generating families. Moreover, any Legendrian with a generating family has an immersed Lagrangian filling with a compatible generating family. These constructions are applied in several directions, in particular to a nonclassical geography question: any graded group satisfying a duality condition can be realized as the generating family homology of a connected Legendrian submanifold in R 2nC1 or in the 1‐jet

Generating family invariants for Legendrian links of unknots

Theory is developed for linear-quadratic at infinity generating families for Legendrian knots in R 3 . It is shown that the unknot with maximal Thurston‐Bennequin invariant of 1 has a unique linear-quadratic at infinity generating family, up to fiber-preserving diffeomorphism and stabilization. From this, invariant generating family polynomials are constructed for 2‐component Legendrian links where each component is a maximal unknot. Techniques are developed to compute these polynomials, and computations are done for two families of Legendrian links: rational links and twist links. The polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links.

Legendrian Circular Helix Links

Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to ℝ n × S n −1 . These links are distinguished by applying the theory of generating functions to contact manifolds.

Obstructions to Lagrangian cobordisms between Legendrians via generating families

The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yield structural information about invariants of Legendrian submanifolds. For example, the obstructions devised in this paper show that there is no generating family compatible Lagrangian cobordism between the Chekanov-Eliashberg Legendrian

Symplectic geometry and topology

m(5_2)

The surgery unknotting number of Legendrian links

knots. Further, the generating family cohomology groups of a Legendrian submanifold restrict the topology of a Lagrangian filling. Structurally, the generating family cohomology of a Legendrian submanifold satisfies a type of Alexander duality that, when the Legendrian is null-cobordant, can be seen as Poincare duality of the associated Lagrangian filling. This duality implies the Arnold Conjecture for Legendrian submanifolds with linear-at-infinity generating families. The results are obtained by developing a generating family version of wrapped Floer cohomology and establishing long exact sequences that arise from viewing the the spaces underlying these cohomology groups as mapping cones.

Symplectic packings in cotangent bundles of tori

Introduction Introduction to symplectic topology Introduction Basics Mosers argument The linear theory The nonsqueezing theorem and capacities Sketch proof of the nonsqueezing theorem Bibliography Holomorphic curves and dynamics in dimension three Problems, basic concepts and overview Analytical tools The Weinstein conjecture in the overtwisted case The Weinstein conjecture in the tight case Some outlook Bibliography An introduction to the Seiberg-Witten equations on symplectic manifolds Introduction Background from differential geometry Spin and the Seiberg-Witten equations The Seiberg-Witten invariants The symplectic case, part I The symplectic case, part II Bibliography Lectures on Floer homology Introduction Symplectic fixed points and Morse theory Fredholm theory Floer homology Gromov compactness and stable maps Multi-valued perturbations Bibliography A tutorial on quantum cohomology Introduction Moduli spaces of stable maps

A LEGENDRIAN STRATIFICATION OF RATIONAL TANGLES

QH^*(G/B)

The minimal length of a Lagrangian cobordism between Legendrians

and quantum Toda lattices Singularity theory Toda lattices and the mirror conjecture Bibliography Euler characteristics and Lagrangian intersections Introduction Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Bibliography Hamiltonian group actions and symplectic reduction Introduction to Hamiltonian group actions The geometry of the moment map Equivariant cohomology and the Cartan model The Duistermaat-Heckman theorem and applications to the cohomology of symplectic quotients Moduli spaces of vector bundles over Riemann surfaces Exercises Bibliography Park City lectures on mechanics, dynamics, and symmetry Introduction Reduction for mechanical systems with symmetry Stability, underwater vehicle dynamics and phases Systems with rolling constraints and locomotion Optimal control and stabilization of balance systems Variational integrators Bibliography.