Lenhard L. Ng
We examine the Legendrian analogue of the topological satellite construction for knots, and deduce some results for specific Legendrian knots and links in standard contact three-space and the solid torus. In particular, we show that the Chekanov-Eliashberg contact homology invariants of Legendrian Whitehead doubles of stabilized knots contain no nonclassical information.
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In this paper we study Legendrian knots in the knot types of satellite knots. In particular, we classify Legendrian Whitehead patterns and learn a great deal about Legendrian braided patterns. We also show how the classification of Legendrian patterns can lead to a classification of the associated satellite knots if the companion knot is Legendrian simple and uniformly thick. This leads to new Legendrian and transverse classification results for knots in the 3-sphere with its standard contact structure as well as a more general perspective on some previous classification results.
We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites by stacking up a sequence of elementary cobordisms. This construction narrows the search for "non-decomposable" Lagrangian cobordisms and yields new families of decomposable Lagrangian slice knots. Finally, we show that the maximum Thurston-Bennequin number of a smoothly slice knot provides an obstruction to any Legendrian satellite of that knot being Lagrang…
We study satellites of Legendrian knots in R^3 and their relation to the Chekanov-Eliashberg differential graded algebra of the knot. In particular, we generalize the well-known correspondence between rulings of a Legendrian knot in R^3 and augmentations of its DGA by showing that the DGA has finite-dimensional representations if and only if there exist certain rulings of satellites of the knot. We derive several consequences of this result, notably that the question of existence of ungraded finite-dimensional representations for the DGA of a Legendrian knot depends only on the topological typ…
Parametric satellites and connected-sums in the space of Legendrian embeddings
本文在Legendrian嵌入空间中引入参数化卫星与连通和构造，生成了高阶同伦新不变量，并构造出具有非平凡LCH monodromy不变量的Legendrian嵌入环的新无限族。
Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory for R^3, Poincare-Chekanov polynomials and characteristic algebras can be associated to such links. The theory is applied to distinguish various knots, as well as links that are closures of Legendrian versions of rational tangles. For a large number of two-component links, the Poincare-Chekanov polynomials agree with the polynomials defined through the theory of generating functions. Examples are given of knots and links which differ by an even number of horiz…
Convex disks with Legendrian boundary in overtwisted contact 3-manifolds
本研究在闭过扭切触3-流形中对带固定特征叶与Legendrian边界的凸圆盘做切触同痕分类，验证其符合h-原理，推导相关Legendrian平凡纽结空间的h-原理性质，并确定过扭3-球面中一类Legendrian纽结补的切触映射类群。
Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact La…
For any Legendrian link, L, in (\R^3, \ker(dz-y\,dx)) we define invariants, Aug_m(L,q), as normalized counts of augmentations from the Legendrian contact homology DGA of L into a finite field of order q where the parameter m is a divisor of twice the rotation number of L. Generalizing a result of Ng and Sabloff for the case q =2, we show the augmentation numbers, Aug_m(L,q), are determined by specializing the m-graded ruling polynomial, R^m_L(z), at z = q^{1/2}-q^{-1/2}. As a corollary, we deduce that the ruling polynomials are determined by the Legendrian contact homology DGA.
On the Legendrian realisation of loops of knots
本研究针对T. Kálmán提出的勒让德嵌入到光滑嵌入基本群同态是否为满射的公开问题，对三大主族无穷多纽结类型及所有稳定化勒让德代表元给出了肯定解答。
We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. The overarching theme of this paper is to show that in many contexts, surgery on transverse knots is more natural than surgery on Legendrian knots. Besides reinterpreting surgery on Legendrian knots in terms of transverse knots, our main results on are in two complementary directions: conditions under which inadmissible transverse surgery (\textit{cf.\@} positive contact surgery on Legendrian knot…
Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that are smoothly the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is "decomposable" into a composition of these elementary building blocks. We will describe these and other combi…
Lagrangian Zigzag Cobordisms
本文研究了标准接触三维空间中Legendrian纽结基于拉格朗日之字形配边的等价关系，将其与光滑配边和拉格朗日配边进行比较，分析了相应度量幺半群的结构，并探讨了其与纽结非经典不变量的联系。
We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of …
Goodman surgery and projectively Anosov flows
本文将Goodman手术推广至射影Anosov流范畴，，证明了任何接触Anosov流均可通过手术构造新流，并在双曲3-流形上生成新例子，同时将双接触手术与经典接触手术关联并推导出紧性结果。
We consider the question of when the operation of contact surgery with positive surgery coefficient, along a knot $K$ in a contact 3-manifold $Y$, gives rise to a weakly fillable contact structure. We show that this happens if and only if $Y$ itself is weakly fillable, and $K$ is isotopic to the boundary of a properly embedded symplectic disk inside a filling of $Y$. Moreover, if $Y'$ is a contact manifold arising from positive contact surgery along $K$, then any filling of $Y'$ is symplectomorphic to the complement of a suitable neighborhood of such a disk in a filling of $Y$. Using this resu…
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