Ichiro Torisu
In this article, we consider the maximal value of the Thurston--Bennequin invariant of Legendrian knots which topologically represent a fixed knot type in the standard contact 3-space and we prove a formula of the value under the connected sum operation of knots.
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We classify Legendrian knots of topological type $7_6$ having maximal Thurston--Bennequin number confirming the corresponding conjectures of Chongchitmate--Ng.
We study the Thurston-Bennequin number of complete and complete bipartite Legendrian graphs. We define a new invariant called the total Thurston-Bennequin number of the graph. We show that this invariant is determined by the Thurston-Bennequin numbers of 3-cycles for complete graphs and by the Thurston-Bennequin number of 4-cycles for complete bipartite graphs. We discuss the consequences of these results for K_4, K_5 and K_{3,3}.
On Legendrian Thurston-Bennequin-symmetrical graphs
本文研究Legendrian图的Thurston-Bennequin不变量计算准则，证明其在有限顶点图中成立，并展示无限例证，揭示图的高对称性。
We show that there exists a Legendrian knot with maximal Thurston-Bennequin invariant whose contact homology is trivial. We also provide another Legendrian knot which has the same knot type and classical invariants but nonvanishing contact homology.
It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the lower degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.
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.
For a knot $K$ the cube number is a knot invariant defined to be the smallest $n$ for which there is a cube diagram of size $n$ for $K$. There is also a Legendrian version of this invariant called the \emph{Legendrian cube number}. We will show that the Legendrian cube number distinguishes the Legendrian left hand torus knots with maximal Thurston-Bennequin number and maximal rotation number from the Legendrian left hand torus knots with maximal Thurston-Bennequin number and minimal rotation number.
We define the Thurston-Bennequin polytope of a two-component link as the convex hull of all pairs of integers that arise as framings of a Legendrian representative. The main result of this paper is a description of the Thurston-Bennequin polytope for two-bridge links. As an application, we construct non-quasipositive surfaces in $\R^3$ all whose sub-annuli are quasipositive.
We give a method for constructing a Legendrian representative of a knot in $S^3$ which realizes its maximal Thurston-Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^4$, and the resulting Legendrian representative is often very complicated (relative to the complexity of the topological knot type). As an application, we construct infinitely many knots in $S^3$ each of which yields a reducible 3-manifold by a Legendrian surgery in the standard tight contact structure. This disproves a conjecture of Lidman and Sivek.
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.
Fundamental generalized Legendrian rack and classical invariants
该研究考察 Legendrian 纽结的基本 GL-rack 同构与其经典不变量之间的关系，证明同构会迫使二者的 Thurston-Bennequin 数和旋转数分别相同或相反。
We establish a relation between several easy-to-calculate numerical invariants of generic knots in $\mathbb{H}^2 \times S^1$, and use it to give a new method of computing the Thurston-Bennequin number of Legendrian knots in the tight contact $\mathbb{R}^3$.
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。
All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $ξ$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight orie…
We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,ξ)$, whenever $ξ$ is tight. More specifically, we show that the self-linking number of a transverse link $T$ in $(M,ξ)$, such that the boundary of its tubular neighbourhood consists of incompressible tori, is bounded by the Thurston norm $||T||_T$ of $T$. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-posit…
We compute the maximal Thurston-Bennequin number for a Legendrian two-bridge knot or oriented two-bridge link in standard contact R^3, by showing that the upper bound given by the Kauffman polynomial is sharp. As an application, we present a table of maximal Thurston-Bennequin numbers for prime knots with nine or fewer crossings.
A flat plumbing basket is a surface consisting a disk and finitely many bands which are contained in distinct pages of the trivial open book decomposition of $\mathbf{S}^{3}$. In this paper, we construct a Legendrian link from a flat plumbing basket, and we describe a relation among the self-linking number, the Thurston-Bennequin number and the flat plumbing basket number of the link. As a corollary, we determine the flat plumbing basket numbers of torus links.
On Legendrian representatives of non-fibered knots
该研究在接触拓扑领域，证明了达到三维Thurston-Bennequin界的非平凡纽结存在tb=0的勒让德代表元，结果可推广至特定接触流形，还给出近纤维化纽结达到该界的条件，填补了3-流形相关研究空白。
The problem of classification of Legendrian knots (links) up to isotopy in the class of Legendrian embeddings (Legendrian isotopy) naturally leads to the following two subproblems. The first of them is: which combinations of the three classical invariants can be realized by a Legendrian knot? (It is well-known that each invariant by itself can be realized by a Legendrian knot). The first step in this direction was done by Bennequin who proved that the Bennequin number of a Legendrian knot is less than twice its genus. The second subproblem is whether there exist a pair of Legendrian knots whic…
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 provide the first example of a Legendrian knot with nonvanishing contact homology whose Thurston-Bennequin invariant is not maximal.
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。
In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{-2n}$ has exactly $\lceil\frac{n^2}2\rceil$ Legendrian representatives with maximal Thurston--Bennequin number, and $\lceil\frac{n}{2}\rceil$ t…
We apply Heegaard-Floer homology theory to establish generalized slicing Bennequin inequalities closely related to a recent result of T. Mrowka and Y. Rollin proved using Seiberg-Witten monopoles.
A first order Vassiliev invariant of an oriented knot in an $S^1$-fibration and a Seifert fibration over a surface is constructed. It takes values in a quotient of the group ring of the first homology group of the total space of the fibration. It gives rise to an invariant of wave fronts on surfaces and orbifolds related to the Bennequin-type invariants of the Legendrian curves studied by F. Aicardi, V. Arnold, M. Polyak, and S. Tabachnikov. Formulas expressing these relations are presented. We also calculate Turaev's shadow for the Legendrian lifting of a wave front. This allows to use in the…
We give a simple unified proof for several disparate bounds on Thurston-Bennequin number for Legendrian knots and self-linking number for transverse knots in R^3, and provide a template for possible future bounds. As an application, we give sufficient conditions for some of these bounds to be sharp.
We prove that two Legendrian knots in a contact structure which is trivializable as a plane bundle are Legendrian isotopic provided that (1) they are isotopic as framed knots, (2) they have the same rotation number with respect to some parallelization of the contact structure, and (3) there is an overtwisted disk disjoint with both knots. (For zero-homologous knots the condition (1) reads as: (1a) they are isotopic as topological knots, and (1b) they have the same Thurston-Bennequin invariant.) Then we discuss the situation when condition (3) is not fulfilled, in particular that of non-loose L…
The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling, therefore its Thurston-Bennequin number is maximal among Legendrian representatives of A. We derive consequences for the Kauffman polynomial and Khovanov homology of +adequate links.
Using a knot concordance invariant from the Heegaard Floer theory of Ozsvath and Szabo, we obtain new bounds for the Thurston-Bennequin and rotation numbers of Legendrian knots in S^3. We also apply these bounds to calculate the knot concordance invariant for certain knots.
In this paper, we introduce a rational $τ$ invariant for rationally null-homologous knots in contact 3-manifolds with nontrivial Ozsváth-Szabó contact invariants. Such an invariant is an upper bound for the sum of rational Thurston-Bennequin invariant and the rational rotation number of the Legendrian representatives of the knot. In the special case of Floer simple knots in L-spaces, we can compute the rational $τ$ invariants by correction terms.
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…
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.
Distinguishing Power of 4-Legendrian Permutation Racks
本文研究4-Legendrian排列架在区分Legendrian结中的能力，证明其无法区分共享相同结类型、Thurston-Bennequin数和旋转数的结，但能恢复这三个经典不变量。
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…
Computing the Thurston-Bennequin invariant in open books
本文提出了一种计算开书页面上Legendrian结Thurston-Bennequin不变量的算法，并扩展到任意3流形中的有理无缠绕结。
In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in $S^1\times S^2$ which is a rational function in variables $a$ and $s$ and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in $S^1\times S^2$ and shows that the invariant is in fact a Laurent polynomial in $a$ and $z= s -s^{-1}$. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legen…
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…
Bounds on Legendrian Double Twist Knots
该研究针对标准紧接触三维球面中的双扭转纽结，利用凸曲面理论的旁路技术与紧接触结构分类，给出K(4,m)族最大瑟斯顿-本内奎恩代表的勒让德同痕类数量上界。
In this paper we will show how to classify Legendrian and transverse knots in the knot type of "sufficiently positive" cables of a knot in terms of the classification of the underlying knot. We will also completely explain the phenomena of "Legendrian large" cables. These are Legendrian representatives of cables that have Thurston-Bennequin invariant larger that the framing coming from the cabling torus. Such examples have only recently, and unexpectedly, been found. We will also give criteria that determines the classification of Legendrian and transverse knots the the knot type of negative c…
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.
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.
In this paper we classify Legendrian and transverse knots in the knot types obtained from positive torus knots by cabling. This classification allows us to demonstrate several new phenomena. Specifically, we show there are knot types that have non-destabilizable Legendrian representatives whose Thurston-Bennequin invariant is arbitrarily far from maximal. We also exhibit Legendrian knots requiring arbitrarily many stabilizations before they become Legendrian isotopic. Similar new phenomena are observed for transverse knots. To achieve these results we define and study "partially thickenable" t…
The Legendrian knot complement problem
研究证明3球面中Legendrian纽结的接触同构类型决定其补集，并通过反例指出该结论不适用于链环，同时提供计算Thurston-Bennequin不变量的新方法。
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…
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.
For any generic immersion of a Petersen graph into a plane, the number of crossing points between two edges of distance one is odd. The sum of the crossing numbers of all $5$-cycles is odd. The sum of the rotation numbers of all $5$-cycles is even. We show analogous results for $6$-cycles, $8$-cycles and $9$-cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a $5$-cycle that is not an unknot with maximal Thurston-Bennequin number, and the sum of all Thurston-Bennequin numbers of the cycles is $7$ times the sum of all Thurston-Bennequin numbers of the $5$-cycles. We …
Examples are given of prime Legendrian knots in the standard contact 3-space that have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny Ng. These are constructed using a new `Legendrian tangle replacement' technique. This technique is then used to show that the phenomenon of multiple Chekanov polynomials is in fact quite common. Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston-Bennequin Legendrian knots for each knot type of nine or fewer cros…
We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$-manifold $(M, ξ)$ which admits an "admissable" open book $(B, f)$ (supporting $ξ$) for $L$. We show that to any such $L$ and a fixed page $X$, one can assign an integer $M\mathcal{P}_{X}(L)$, called "Relative Maximal Page Crossing Number of $L$ with respect to $X$", which is invariant under Legendrian isotopies of $L$. We also show that one can extend this to a page-free invariant, i.e., one can assign an integer $M\mathcal{P}_{(B,f)}(L)$, called "Absolute Maximal P…
We study Legendrian knots in a cabled knot type. Specifically, given a topological knot type K, we analyze the Legendrian knots in knot types obtained from K by cabling, in terms of Legendrian knots in the knot type K. As a corollary of this analysis, we show that the (2,3)-cable of the (2,3)-torus knot is not transversely simple and moreover classify the transverse knots in this knot type. This is the first classification of transverse knots in a non-transversely-simple knot type. We also classify Legendrian knots in this knot type and exhibit the first example of a Legendrian knot that does …
We prove a finiteness property of the values of the skein polynomial of homogeneous knots which allows to establish large classes of such knots to have arbitrarily unsharp Bennequin inequality (for the Thurston-Bennequin invariant of any of their Legendrian embeddings in the standard contact structure of R^3), and a give a short proof that there are only finitely many among these knots that have given genus and given braid index.
In this short note we observe that a result of Eliashberg and Polterovitch allows to use the doubly slice genus as an obstruction for a Legendrian knot to be a slice of a concordance from the trivial Legendrian knot with maximal Thurston-Bennequin invariant to itself. This allows to obstruct concordances from the Pretzel knot P(3,-3,-m) when m > 3 to the unknot. Those examples are of interest because the Legendrian contact homology algebra cannot be used to obstruct such a concordance.
In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. We give an example of a tight confoliation $ξ$ on $T^3$ violating the Thurston-Bennequin inequalities. This answers a question from \cite{confol} negatively. Although the tightness of a confoliation does not imply the Thurston-Bennequin inequalities, it is still possible to prove restrictions on homotopy classes of plane fields which contain tight confoliations. The failure of the Thurston-Bennequin inequalities for tight confoliations is due to the presence of overtwisted stars. Overtwisted stars are part…
We use parameterized Morse theory on the pages of an open book decomposition to efficiently encode the contact topology in terms of a labelled graph on a disjoint union of tori (one per binding component). This construction allows us to generalize the notion of the front projection of a Legendrian knot from the standard contact $\mathbb{R}^3$ to arbitrary closed contact $3$-manifolds. We describe a complete set of moves on such front diagrams, extending the standard Legendrian Reidemeister moves, and we give a combinatorial formula to compute the Thurston-Bennequin number of a nullhomologous L…
In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using these examples, we are able to provide a new proof of the local Tho…
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。
On a generalization of Jones polynomial and its categorification for Legendrian Knots
本文通过新型 skein 关系推广 Jones 多项式，并构造相应的 Khovanov 型范畴化，用于刻画 Legendrian 纽结。
If a Legendrian knot $Λ$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $Σ$ in the standard symplectic 4-ball, then the genus of $Σ$ is equal to the slice genus of (the smooth knot underlying) $Λ$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $Σ$ is zero as well as the rotation number of $Λ$, and moreover, the linearized contact homology of $Λ$ with respect to the augmentation induced by $Σ$ is isomorphic to the (singular) homology of $Σ$. It was asked in arXiv:1212.1519 whether the converse of this statement is true. We give a n…
We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we show that the strongly non-loose transverse realizations $T$ are classified by their non-zero invariants $\mathfrak T(T)$ in the minus version of the knot Floer homology. However, not all the elements of $HFK^-(T_{(p,q)})$ can be realized. Along the way, w…
Tight contact structures on some families of small Seifert fiber spaces
本文提出一种通过比较等价 Thurston–Bennequin 标架的 Legendrian 表示来证明手术后接触结构仍不同的方法，并结合凸曲面理论分类若干小 Seifert 纤维空间族上的紧接触结构。
Using the knot Floer homology filtration, we define invariants associated to a knot in a three-manifold possessing non-vanishing Floer co(homology) classes. In the case of the Ozsvath-Szabo contact invariant we obtain an invariant of knots in a contact three-manifold. This invariant provides an upper bound for the Thurston-Bennequin plus rotation number of any Legendrian realization of the knot. We use it to demonstrate the first systematic construction of prime knots in contact manifolds other than the three-sphere with negative maximal Thurston-Bennequin invariant. Perhaps more interesting, …
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