Klaus Mohnke
We use an estimate on the Thurston--Bennequin invariant of a Legendrian link in terms of its Kauffman-polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian links of Legendrian unknots.
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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 N…
Hard Legendrian unknots
本文研究Legendrian未结直线的Reidemeister难度，通过正常规则阻碍了多个无限家族的硬未结图无法用最大tb未结前图绘制，并界定了部分硬未结图的最小可能扭度。
We classify Legendrian rational unknots with tight complements in the lens spaces L(p,1) up to coarse equivalence. As an example of the general case, this classification is also worked out for L(5,2). The knots are described explicitly in a contact surgery diagram of the corresponding lens space.
The contact mapping class group and rational unknots in lens spaces
本文通过单参数凸曲面理论分类透镜空间紧接触结构中的勒让德与横截有理解结，进而确定了标准接触结构的接触映射类群。
Singular Legendrian unknot links and relative Ginzburg algebras
本文通过构造与箭图及子箭图相关的带停止Weinstein流形，证明其奇异勒让德平凡结链的Chekanov--Eliashberg dg-代数与相对Ginzburg代数拟同构，并由此推出两者具有强相对光滑Calabi--Yau结构。
It is shown that Legendrian (resp. transverse) cable links in the 3-sphere with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston-Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space of the circle with its standard tight contact structure.
Constructions and Isotopies of High-Dimensional Legendrian Spheres
本文研究任意维接触流形中勒让德球的三种构造，证明它们均与勒让德平凡结同痕，推广了Courte和Ekholm的结果。
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 polyno…
We investigate the question of the existence of a Lagrangian concordance between two Legendrian knots in $\mathbb{R}^3$. In particular, we give obstructions to a concordance from an arbitrary knot to the standard Legendrian unknot, in terms of normal rulings. We also place strong restrictions on knots that have concordances both to and from the unknot and construct an infinite family of knots with non-reversible concordances from the unknot. Finally, we use our obstructions to present a complete list of knots with up to 14 crossings that have Legendrian representatives that are Lagrangian slic…
We classify Legendrian unknots in overtwisted contact structures on $S^3$. In particular, we show that up to contact isotopy for every pair $(n,\pm(n-1))$ with $n>0$ there are exactly two oriented non-loose Legendrian unknots in $S^3$ with Thurston-Bennequin invariant $n$ and rotation number $\pm(n-1)$. (Only one overtwisted contact structure on $S^3$ admits a non-loose unknot $K$ and the classical invariants have to be $\mathrm{tb}(K)=n$ and $\mathrm{rot}(K)=\pm(n-1)$ for $n>1$.) This can be used to prove two results attributed to Y.~Che\-kan\-ov: The first implies that the contact mapping cl…
On contact cosmetic surgery
本文证明接触整容手术猜想对所有非平凡Legendrian结成立，但可能排除Lagrangian切片结，并讨论Legendrian平凡结的接触整容手术。
We show that any knot which is smoothly the closure of a 3-braid cannot be Lagrangian concordant to and from the maximum Thurston-Bennequin Legendrian unknot except the unknot itself. Our obstruction comes from drawing the Weinstein handlebody diagrams of particular symplectic fillings of cyclic branched double covers of knots in $S^3$. We use the Legendrian contact homology differential graded algebra of the links in these diagrams to compute the symplectic homology of these fillings to derive a contradiction. As a corollary, we find an infinite family of contact manifolds which are rational …
Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds
该研究通过构造非简单同胚类的Legendrian链接，展示了在过扭接触结构中，Legendrian同胚与框架同胚、同伦及链接同伦之间的差异，揭示了非简单性现象在相同光滑同胚类中的共存。
Characterizing slopes for Legendrian knots
该研究在勒让德纽结同痕类由经典不变量唯一确定的假设下，建立了光滑与接触特征斜率的关联准则，并将其应用于多种纽结的勒让德代表的特征斜率研究。
We study Legendrian singular links up to contact isotopy. Using a special property of the singular points, we define the singular connected sum of Legendrian singular links. This concept is a generalization of the connected sum and can be interpreted as a tangle replacement, which provides a way to classify Legendrian singular links. Moreover, we investigate several phenomena only occur in the Legendrian setup.
Legendrian Torus and Cable Links
本文对Legendrian环面链环及均匀粗纽结类型的缆绳链环进行分类，首次发现无穷族Legendrian链环中光滑对称性不可由Legendrian同痕实现，并给出非可去稳定化但非最大Thurston-Bennequin不变量的例子。
Legendrian Lavrentiev links
针对拉夫连季耶夫曲线，定义了勒让德拉夫连季耶夫曲线及同痕，证明其链环的勒让德同痕等价类与光滑类一致，为相关曲线等价性研究提供基础。
Lisa Traynor has described an example of a two-component Legendrian `circular helix link' in the 1-jet space of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to that same link with the order of the two components reversed. We give a complete classification of the Legendrian realisations of this topological link type, as well as all other `cable links' in that 1-jet space.
On Ruling polynomials of Legendrian links
该研究刻画了勒让德链的分次与非分次统治多项式，构造了实现对应多项式的勒让德链，其增广簇具平凡簇代数结构，还构造了满足特定条件的勒让德纽结。
Legendrian Hopf links
本文在任意接触结构的三维球面中，完全分类了 Hopf 链环的 Legendre 实现，分类标准为粗等价。
Strongly exceptional Legendrian connected sum of two Hopf links
本文首次对两个霍夫链接连和的强异常Legendrian实现进行了完整粗分类，揭示了其在接触3球面中的结构特性。
In this paper, we study contact surgeries along Legendrian links in the standard contact 3-sphere. On one hand, we use algebraic methods to prove the vanishing of the contact Ozsváth-Szabó invariant for contact $(+1)$-surgery along certain Legendrian two-component links. The main tool is a link surgery formula for Heegaard Floer homology developed by Manolescu and Ozsváth. On the other hand, we use contact-geometric argument to show the overtwistedness of the contact 3-manifolds obtained by contact $(+1)$-surgeries along Legendrian two-component links whose two components are linked in some sp…
Legendrian Hopf links in L(p,1)
本文在透镜空间L(p,1)中，对任意接触结构下的Hopf链环的Legendrian实现进行了粗等价分类。
We show that a link in an open book can be realized as a strongly quasipositive braid if and only if it bounds a Legendrian ribbon with respect to the associated contact structure. This generalizes a result due to Baader and Ishikawa for links in the three-sphere. We highlight some related techniques for determining whether or not a link is strongly quasipositive, emphasizing applications to fibered links and satellites.
Spaces of Legendrian cables and Seifert fibered links
本文确定了标准接触3-球面中具最大瑟斯顿-本内昆不变量的若干勒让德纽结和链环嵌入空间的同伦型，给出充分正缆线的递归公式，证明相关接触结构空间的同伦等价性与可缩性，还发现满足单射h-原理的无穷多新分支。
We use the spanning tree model for Khovanov homology to study Legendrian links. This leads to an alternative proof for Ng's Khovanov bound for the Thurston-Bennequin number and to both a necessary and a sufficient condition for this bound to be sharp.
We study an $A_\infty$ category associated to Legendrian links in $\mathbb{R}^3$ whose objects are $n$-dimensional representations of the Chekanov-Eliashberg differential graded algebra of the link. This representation category generalizes the positive augmentation category and we conjecture that it is equivalent to a category of sheaves of microlocal rank $n$ constructed by Shende, Treumann, and Zaslow. We establish the cohomological version of this conjecture for a family of Legendrian $(2,m)$ torus links.
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…
Non-loose Legendrian Hopf links in lens spaces
本文给出了透镜空间中非宽松 Legendrian Hopf 链环的完整分类，推广了先前结果，并提供了无 Giroux 扭转情形的接触手术图算法。
We introduce a notion of cardinality for the augmentation category associated to a Legendrian knot or link in standard contact R^3. This `homotopy cardinality' is an invariant of the category and allows for a weighted count of augmentations, which we prove to be determined by the ruling polynomial of the link. We present an application to the augmentation category of doubly Lagrangian slice knots.
Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links (up to a normalized factor) \cite{HR15}. In this article, we generalize this picture to Legendrian tangles, which are morally the pieces obtained by cutting Legendrian link fronts along 2 vertical lines. Moreover, we show that the Ruling polynomials for Legendrian tangles satisfy the composition axiom. In the special case of Legendrian knots, our arguments pr…
An elementary stabilization of a Legendrian link $L$ in the spherical cotangent bundle $ST^*M$ of a surface $M$ is a surgery that results in attaching a handle to $M$ along two discs away from the image in $M$ of the projection of the link $L$. A virtual Legendrian isotopy is a composition of stabilizations, destabilizations and Legendrian isotopies. In contrast to Legendrian knots, virtual Legendrian knots enjoy the property that there is a bijective correspondence between the virtual Legendrian knots and the equivalence classes of Gauss diagrams. We study virtual Legendrian isotopy classes o…
Cable links of uniformly thick knot types
本文基于Legendrian手术引入新方法，研究了均匀粗细但非Legendrian简单结型缆链的Legendrian实现，发现稳定化链的新同痕现象，并对充分负斜率缆链及twist knots的负缆进行了分类。
Given a Legendrian link in $\#^k(S^1\times S^2)$, we extend the definition of a normal ruling from $J^1(S^1)$ given by Lavrov and Rutherford and show that the existence of an augmentation to any field of the Chekanov-Eliashberg differential graded algebra over $\mathbb{Z}[t,t^{-1}]$ 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. We show a similar correspondence …
In this article, we introduce rack invariants of oriented Legendrian knots in the 3-dimensional Euclidean space endowed with the standard contact structure, which we call Legendrian racks. These invariants form a generalization of the quandle invariants of knots. These rack invariants do not result in a complete invariant, but detect some of the geometric properties such as cusps in a Legendrian knot. In the case of topologically trivial Legendrian knots, we test this family of invariants for its strengths and limitations. We further prove that these invariants form a natural generalization of…
We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar methods a result involving the upper bound given by the HOMFLY polynomial and 2-graded rulings is proved.
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…
For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán 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 $\mathbb{Z}_2[H_1(L)]$ and distinguish the resulting augmentations.
We present the first examples of elements in the fundamental group of the space of Legendrian links in the standard contact 3-sphere whose action on the Legendrian contact DGA is of infinite order. This allows us to construct the first families of Legendrian links that can be shown to admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These families include the first known Legendrian links with infinitely many fillings that are not rainbow closures of positive braids, and the smallest Legendrian link with infinitely many fillings known to date. We discuss how to use our e…
For Legendrian links in the 1-jet space of $S^1$ we show that the 1-graded ruling polynomial may be recovered from the Kauffman skein module. For such links a generalization of the notion of normal ruling is introduced. We show that the existence of such a generalized normal ruling is equivalent to sharpness of the Kauffman polynomial estimate for the Thurston-Bennequin number as well as to the existence of an ungraded augmentation of the Chekanov-Eliashberg DGA. Parallel results involving the HOMFLY-PT polynomial and 2-graded generalized normal rulings are established.
We show that for any Legendrian link $L$ in the $1$-jet space of $S^1$ the $2$-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the $2$-graded case, we are able to use $0$-…
We establish an upper bound for the Thurston-Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer crossings.
In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the Bennequin bound is sharp for fibered knot types. Finally we study rational unknots and show they are weakly Legendrian and transversely simple. This version of the paper corrects the definition of rational self-linking number in the previous and published version of the paper. Wi…
In this paper, we show that any topological knot or link in $S^1 \times S^2$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $S^1 \times S^2$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces $L(p,1)$.
Generalised Legendrian racks of Legendrian links
本文提出广义Legendrian rack（GL-rack）作为有向Legendrian链环的纯rack理论不变量，证明其能区分无穷多Legendrian平凡纽结与三叶纽结，并揭示其代数结构与模范畴等价性。
We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of affine type $\tilde{\mathsf{D}} \tilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with certain symmetries as seeds of type $\tilde{\mathsf{B}}_n$, $\tilde{\mathsf{F}}_4$, $\tilde{\mathsf{G}}_2$, and $\mathsf{E}_6^{(2)}$. These families are the first known Legendrian links with infinitely many fillings that exhaust all seeds in the corresponding cluster structures. Furthermore, we show that Legendrian realization of Coxeter mutation of type $\tilde{\mathsf{D}}$ corresponds…
The Thurston-Bennequin invariant provides one notion of self-linking for any homologically-trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in Euclidean space. Our definition is based upon a reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.
We prove that there are at least seeds many exact embedded Lagrangian fillings for Legendrian links of type $\mathsf{ADE}$. We also provide seeds many Lagrangian fillings with certain symmetries for type $\mathsf{BCFG}$. Our main tools are $N$-graphs and the combinatorics of seed patterns of finite type.
Motion groups, Legendrian links, and non-trivial loops of embeddings
该研究定义了接触3-流形中勒让德链的勒让德运动群，证明其同构于未参数化勒让德嵌入空间的基本群，还刻画了标准接触3-球面中勒让德纽结的运动，并在附录中对比了相关光滑情形技术与达姆同态。
Legendrian links and déjà vu moments
本文提出似曾相识勒让德链环的概念，刻画分量可由正勒让德同痕连接但同痕不可嵌入的情形，并在多种几何背景中构造实例并研究其基本性质。
Conjectures about virtual Legendrian knots and links
本文提出并部分证明了关于virtual Legendrian knots的猜想，包括其与普通Legendrian knots的关系、与广义时空因果性的关联，并在2维M和(2+1)维时空中证明了这些猜想及Arnold的4尖点猜想版本。
We consider S^1-families of Legendrian knots in the standard contact R^3. We define the monodromy of such a loop, which is an automorphism of the Chekanov-Eliashberg contact homology of the starting (and ending) point. We prove this monodromy is a homotopy invariant of the loop. We also establish techniques to address the issue of Reidemeister moves of Lagrangian projections of Legendrian links. As an application, we exhibit a loop of right-handed Legendrian torus knots which is non-contractible in the space Leg(S^1,R^3) of Legendrian knots, although it is contractible in the space Emb(S^1,R^3…
An L-infinity structure for Legendrian contact homology
本文为三维空间中的 Legendre 纽结或链环构造了扩展 Chekanov-Eliashberg 微分分次代数的 L-无穷代数，并由此在交换 Legendre 接触同调上建立 Poisson 括号，证明其是合痕不变量。
Strongly Invertible Legendrian Links
本文在标准三维接触空间中引入强可逆勒让德链环，建立等变类比基本结果，并定义极大等变瑟斯顿-贝内金数，展示无穷多一致性实例，同时提出不一致猜想。
In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we determine when each prime 2-string tangle with up to seven crossings embeds into the unknot, the unlink or a split link.
Upper bounds for the Lagrangian cobordism relation on Legendrian links
本文证明了在紧接触3-流形中，具有公共旋转数的零调Legendrian链有限集合关于Lagrangian配边预序存在上界，并据此定义了极小Lagrangian亏格。
Using the combinatorial approach to knot Floer homology, we define an invariant for Legendrian knots in the three-sphere, which takes values in link Floer homology. This invariant can be used to also construct an invariant of transverse knots.
Derivative links in contact topology
本文把R-链环和导数链环理论引入接触拓扑，用于刻画Lagrangian与辛可切片性，并证明Thurston-Bennequin不变量为-1的Legendrian结的切片性等价于存在紧的横截或Legendrian R-链环导数。
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.
We strengthen the link between holomorphic and generating-function invariants of Legendrian knots by establishing a formula relating the number of augmentations of a knot's contact homology to the complete ruling invariant of Chekanov and Pushkar.
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