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Joshua M. Sabloff

Publications and source records attributed to Joshua M. Sabloff.

27 records · Page 2Linked to original sources
arXiv2010

Legendrian Contact Homology in Seifert Fibered Spaces

We define a differential graded algebra associated to Legendrian knots in Seifert fibered spaces with transverse contact structures. This construction is distinguished from other combinatorial realizations of contact homology invariants by the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology is torsion and whose underlying topological knot types are isotopic; such examples exist in any sufficiently complicated contact Seifert fibered space.

math.SG↗
arXiv2009

A Duality Exact Sequence for Legendrian Contact Homology

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact homology CH_* which maps to singular homology. In particular, the sequence implies a duality between the kernel of the map (CH_*\to H_*) and the cokernel of the map (H_* \to CH^*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a in CH_* maps to a class αin CH^* such that α(a)=1. The exact sequence generalizes the duality for Legendrian knots in Euclidean 3-space [24] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [6].

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arXiv2009

Product Structures for Legendrian Contact Homology

Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while preserving computability, we introduce invariant cup and Massey products - and, more generally, an A_\infty structure - on the linearized LCH. We apply the products and A_\infty structure in three ways: to find infinite families of Legendrian knots that are not isotopic to their Legendrian mirrors, to reinterpret the duality theorem of the fourth author in terms of the cup product, and to recover higher-order linearizations of the LCH.

math.SG↗
arXiv2008

Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

Capacities that provide both qualitative and quantitative obstructions to the existence of a Lagrangian cobordism between two $(n-1)$-dimensional submanifolds in parallel hyperplanes of $\mathbb{R}^{2n}$ are defined using the theory of generating families. Qualitatively, these capacities show that, for example, in $\mathbb R^4$ there is no Lagrangian cobordism between two $\infty$-shaped curves with a negative crossing when the lower end is "smaller". Quantitatively, when the boundary of a Lagrangian ball lies in a hyperplane of $\mathbb{R}^{2n}$, the capacity of the boundary gives a restriction on the size of a rectangular cylinder into which the Lagrangian ball can be squeezed.

math.SG↗
arXiv2008

A Partial Ordering on Slices of Planar Lagrangians

A collection of simple closed curves in $\rr^3$ is called a negative slice if it is the intersection of a flat-at-infinity planar Lagrangian surface and $\{y_2 = a \}$ for some $a < 0$. Examples and non-examples of negative slices are given. Embedded Lagrange cobordisms define a relation on slices and in some (and perhaps all) cases this relation defines a partial order. The set of slices is a commutative monoid and the additive structure has an interesting relationship with the ordering relation.

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arXiv2005

Augmentations and Rulings of Legendrian Knots

A connection between holomorphic and generating family invariants of Legendrian knots is established; namely, that the existence of a ruling (or decomposition) of a Legendrian knot is equivalent to the existence of an augmentation of its contact homology. This result was obtained independently and using different methods by Fuchs and Ishkhanov. Close examination of the proof yields an algorithm for constructing a ruling given an augmentation. Finally, a condition for the existence of an augmentation in terms of the rotation number is obtained.

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arXiv2003

Invariants of Legendrian Knots and Coherent Orientations

We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R^3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea of ``coherent orientations'' from the Contact Homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over Z[t,t^{-1}] with a full Z grading.

math.SG↗
arXiv2002

Invariants of Legendrian knots in circle bundles

Let E be a circle bundle over a Riemann surface that supports a contact structure transverse to the fibers. This paper presents a combinatorial definition of a differential graded algebra (DGA) that is an invariant of Legendrian knots in E. The invariant generalizes Chekanov's combinatorial DGA invariant of Legendrian knots in the standard contact 3-space using ideas from Eliashberg, Givental, and Hofer's contact homology. The main difficulty lies in dealing with what are ostensibly 1-parameter families of generators for the DGA; these are solved using ``Morse-Bott'' techniques. As an application, the invariant is used to distinguish two Legendrian knots that are smoothly isotopic, realize a non-trivial homology class, but are not Legendrian isotopic.

math.SG↗