Browsing by Author "Ng, Lenhard L"
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Item Open Access Augmentations and exact Lagrangian cobordisms(2017) Pan, YUTo a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category.
An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots.
We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends.
As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and
find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.
As a related project, we study exact Lagrangian fillings of Legendrian $(2,n)$ links.
For a Legendrian $(2,n)$ torus knot or link with maximal Thurston--Bennequin number,
Ekholm, Honda, and K{\'a}lm{\'a}n constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$--th Catalan number.
We show that these exact Lagrangian fillings are pairwise non--isotopic through exact Lagrangian isotopy.
To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to $\mathbbZ_2[H_1(L)]$ and distinguish the resulting augmentations.
Item Open Access Augmentations and Rulings of Legendrian Links(2016) Leverson, Caitlin JuneFor any Legendrian knot in R^3 with the standard contact structure, we show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a normal ruling of the front diagram, generalizing results of Fuchs, Ishkhanov, and Sabloff. We also show that any even graded augmentation must send t to -1.
We extend the definition of a normal ruling from J^1(S^1) given by Lavrov and Rutherford to a normal ruling for Legendrian links in #^k(S^1\times S^2). We then show that for Legendrian links in J^1(S^1) and #^k(S^1\times S^2), the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over Z[t,t^{-1}] is equivalent to the existence of a normal ruling of the front diagram. For Legendrian knots, we also show that any even graded augmentation must send t to -1. We use the correspondence to give nonvanishing results for the symplectic homology of certain Weinstein 4-manifolds.
Item Open Access Categorification of quantum sl_3 projectors and the sl_3 Reshetikhin-Turaev invariant of framed tangles(2012) Rose, David Emile VatcherQuantum sl_3 projectors are morphisms in Kuperberg's sl_3 spider, a diagrammatically defined category equivalent to the full pivotal subcategory of the category of (type 1) finite-dimensional representations of the quantum group U_q (sl_3 ) generated by the defining representation, which correspond to projection onto (and then inclusion from) the highest weight irreducible summand. These morphisms are interesting from a topological viewpoint as they allow the combinatorial formulation of the sl_3 tangle invariant (in which tangle components are labelled by the defining representation) to be extended to a combinatorial formulation of the invariant in which components are labelled by arbitrary finite-dimensional irreducible representations. They also allow for a combinatorial description of the SU(3) Witten-Reshetikhin-Turaev 3-manifold invariant.
There exists a categorification of the sl_3 spider, due to Morrison and Nieh, which is the natural setting for Khovanov's sl_3 link homology theory and its extension to tangles. An obvious question is whether there exist objects in this categorification which categorify the sl_3 projectors.
In this dissertation, we show that there indeed exist such "categorified projectors," constructing them as the stable limit of the complexes assigned to k-twist torus braids (suitably shifted). These complexes satisfy categorified versions of the defining relations of the (decategorified) sl_3 projectors and map to them upon taking the Grothendieck group. We use these categorified projectors to extend sl_3 Khovanov homology to a homology theory for framed links with components labeled by arbitrary finite-dimensional irreducible representations of sl_3 .
Item Embargo DGA maps Induced by Decomposable Fillings with Z-coefficients(2023) Mohanakumar, ChinduTo every Legendrian link in R3, we can assign a differential graded algebra (DGA) called the Chekanov-Eliashberg DGA. An exact Lagrangian cobordism between two Legendrian links induces a DGA map between the corresponding Chekanov-Eliashberg DGAs, and this association is functorial. This DGA map was written down explicity for exact, decomposable Lagrangian fillings as Z_2-count of certain pseudoholomorphic disks by Ekholm, Honda, and K ́alm ́an, and this was combinatorially upgraded to an integral count by Casals and Ng. However, this upgrade only assigned an automorphism class of DGA maps. We approach the same problem of integral lifts by a different strategy, first done for the differential in the Chekanov-Eliashberg DGA by Ekholm, Etnyre, and Sullivan. Here, we find the precise DGA maps for all exact, decomposable Lagrangian cobordisms through this more analytic method.
Item Open Access Exact Lagrangian Fillings of Legendrian links and Weinstein 4-manifolds(2021) Capovilla-Searle, OrsolaOne approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of Legendrian links in the standard contact 3-sphere, and one can use Floer theoretic invariants to study such fillings. In this thesis we prove that a family of oriented Legendrian links has infinitely many distinct exact orientable Lagrangian fillings which are smoothly isotopic but not smoothly isotopic. To distinguish these fillings we use Floer theoretic techniques developed by Casals and Ng. We provide one of the first examples of a Legendrian link that admits infinitely many planar exact Lagrangian fillings. As part of a collaboration, we also explore obstructions to the existence of exact Lagrangian cobordisms between Legendrian links that can be applied to obstructing certain immersed exact Lagrangian fillings.Weinstein domains are examples of a symplectic manifold with contact boundary that have a handle decomposition compatible with the symplectic structure of the manifold. Weinstein $4$-dimensional domains can be represented with Weinstein handlebody diagrams of Legendrian links in $(\#^m(S^1\times S^2), \xi_{std})$ or $(S^3, \xi_{std}).$ Studying the symplectic topology of Weinstein domains has allowed for new perspectives when studying various manifolds including complex affine varieties. We study the Milnor fibers $M_f$ of isolated unimodular singularities. Keating constructed an exact Lagrangian torus in $M_f$. We show that there are exact infinitely many Hamiltonian non-isotopic Lagrangian tori in $M_f$ using Weinstein handlebody diagrams and exact Lagrangian fillings of Legendrian links. We also show that $M_f$ contains a new infinite set of symplectically knotted Lagrangian spheres. Additionally, we provide a generalization of a criterion for when the symplectic homology of a Weinstein $4$ manifold is non-vanishing given a Weinstein handlebody diagram. Finally, we provide a summary of a second collaboration which studies the complement of smoothings of toric divisors in toric $4$-manifolds. We show that for certain smoothings, these complements have a particular Weinstein structure, and we provide an algorithm to construct the Weinstein handlebody diagram of such complements.
Item Open Access Legendrian contact homology in R^3Etnyre, John B; Ng, Lenhard LThis is an introduction to Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, with a focus on the setting of Legendrian knots in $\mathbb{R}^3$.Item Open Access The Spherical Manifold Realization Problem(2020-05-09) Davis, A. BlytheThe Lickorish-Wallace theorem states that any closed, orientable, connected 3-manifold can be obtained by integral Dehn surgery on a link in S^3. The spherical manifold realization problem asks which spherical manifolds (i.e., those with finite fundamental groups) can be obtained through integral surgery on a knot in S^3. The problem has previously been solved by Greene and Ballinger et al. for lens space and prism manifolds, respectively. In this project, we determine which of the remaining three types of spherical manifolds (tetrahedral, octahedral, and icosahedral) can be obtained by positive integral surgery on a knot in S^3. We follow methods inspired by those presented by Greene.Item Open Access Transverse Homology and Transverse Nonsimplicity(2019) Lian, ChesterWe show that if a transversely nonsimple knot has two braid representatives, related by a negative flype, that can be distinguished by augmentation numbers of transverse homology, then an infinite family of transversely nonsimple knots with the same property can be constructed explicitly. As an application, we give an example of an infinite family of knots that can be proven to be transversely nonsimple by transverse homology, but not by the theta-hat invariant from knot Floer homology. In addition, we exhibit an infinite family of transversely nonsimple knots all of whose mirror images are transversely nonsimple as well; moreover, we prove that if a knot with the aforementioned property is also thin, then the theta-hat invariant must be trivial for all transverse representatives of the knot or its mirror image. Finally, a question posed by Plamenevskaya is as follows: does there exist a right-veering link for which the theta-hat invariant vanishes? A weaker version of this question asks if there exists a nondestabilizable right-veering braid for which the theta-hat invariant vanishes. We answer the latter in the affirmative by providing an example.