64 research outputs found
Non-Orientable Lagrangian Cobordisms between Legendrian Knots
In the symplectization of standard contact -space, , it is known that an orientable Lagrangian cobordism between a
Legendrian knot and itself, also known as an orientable Lagrangian
endocobordism for the Legendrian knot, must have genus . We show that any
Legendrian knot has a non-orientable Lagrangian endocobordism, and that the
crosscap genus of such a non-orientable Lagrangian endocobordism must be a
positive multiple of . The more restrictive exact, non-orientable Lagrangian
endocobordisms do not exist for any exactly fillable Legendrian knot but do
exist for any stabilized Legendrian knot. Moreover, the relation defined by
exact, non-orientable Lagrangian cobordism on the set of stabilized Legendrian
knots is symmetric and defines an equivalence relation, a contrast to the
non-symmetric relation defined by orientable Lagrangian cobordisms.Comment: 23 pages, 18 figure
The Surgery Unknotting Number of Legendrian Links
The surgery unknotting number of a Legendrian link is defined as the minimal
number of particular oriented surgeries that are required to convert the link
into a Legendrian unknot. Lower bounds for the surgery unknotting number are
given in terms of classical invariants of the Legendrian link. The surgery
unknotting number is calculated for every Legendrian link that is topologically
a twist knot or a torus link and for every positive, Legendrian rational link.
In addition, the surgery unknotting number is calculated for every Legendrian
knot in the Legendrian knot atlas of Chongchitmate and Ng whose underlying
smooth knot has crossing number 7 or less. In all these calculations, as long
as the Legendrian link of components is not topologically a slice knot, its
surgery unknotting number is equal to the sum of and twice the smooth
4-ball genus of the underlying smooth link.Comment: 26 pages, 27 figure
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 x Sn-1. These links are distinguished by applying the theory of generating functions to contact manifolds
Lagrangian Cobordisms via Generating Families: Constructions and Geography
Embedded Lagrangian cobordisms between Legendrian submanifolds are produced
from 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
non-classical geography question: any graded group satisfying a duality
condition can be realized as the generating family homology of a connected
Legendrian submanifold in R^{2n+1} or in the 1-jet space of any compact
n-manifold with n at least 2.Comment: 34 pages, 11 figures. v2: corrected a referenc
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
defined 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 refined version of Chekanov's first order polynomials which
are defined via the theory of holomorphic curves.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol5/paper23.abs.htm
The Relative Gromov Width of Lagrangian Cobordisms between Legendrians
We obtain upper and lower bounds for the relative Gromov width of Lagrangian cobordisms between Legendrian submanifolds. Upper bounds arise from the existence of J-holomorphic disks with boundary on the Lagrangian cobordism that pass through the center of a given symplectically embedded ball. The areas of these disks - and hence the sizes of these balls - are controlled by a real-valued fundamental capacity, a quantity derived from the algebraic structure of filtered linearized Legendrian Contact Homology of the Legendrian at the top of the cobordism. Lower bounds come from explicit constructions that use neighborhoods of Reeb chords in the Legendrian ends. We also study relationships between the relative Gromov width and another quantitative measurement, the length of a cobordism between two Legendrian submanifolds
The Minimal Length of the Lagrangian Cobordism Between Legendrians
To investigate the rigidity and flexibility of Lagrangian cobordisms between Legendrian submanifolds, we study the minimal length of such a cobordism, which is a 1-dimensional measurement of the non-cylindrical portion of the cobordism. Our primary tool is a set of real-valued capacities for a Legendrian submanifold, which are derived from a filtered version of Legendrian contact homology. Relationships between capacities of Legendrians at the ends of a Lagrangian cobordism yield lower bounds on the length of the cobordism. We apply the capacities to Lagrangian cobordisms realizing vertical dilations (which may be arbitrarily short) and contractions (whose lengths are bounded below). We also study the interaction between length and the linking of multiple cobordisms as well as the lengths of cobordisms derived from non-trivial loops of Legendrian isotopies
The Relative Gromov Width of Lagrangian Cobordisms between Legendrians
We obtain upper and lower bounds for the relative Gromov width of Lagrangian cobordisms between Legendrian submanifolds. Upper bounds arise from the existence of J-holomorphic disks with boundary on the Lagrangian cobordism that pass through the center of a given symplectically embedded ball. The areas of these disks - and hence the sizes of these balls - are controlled by a real-valued fundamental capacity, a quantity derived from the algebraic structure of filtered linearized Legendrian Contact Homology of the Legendrian at the top of the cobordism. Lower bounds come from explicit constructions that use neighborhoods of Reeb chords in the Legendrian ends. We also study relationships between the relative Gromov width and another quantitative measurement, the length of a cobordism between two Legendrian submanifolds
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