Legendrian Links Research Articles

HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links

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-graded ruling polynomials to distinguish many homotopically non-trivial Legendrian links with identical classical invariants.

Legendrian Links, Causality, and the Low Conjecture

Let $(X^{m+1}, g)$ be a globally hyperbolic spacetime with Cauchy surface diffeomorphic to an open subset of $\mathbb R^m$. The Legendrian Low conjecture formulated by Nat\'ario and Tod says that two events $x,y\in\ss$ are causally related if and only if the Legendrian link of spheres $\mathfrak S_x, \mathfrak S_y$ whose points are light geodesics passing through $x$ and $y$ is non-trivial in the contact manifold of all light geodesics in $X$. The Low conjecture says that for $m=2$ the events $x,y$ are causally related if and only if $\mathfrak S_x, \mathfrak S_y$ is non-trivial as a topological link. We prove the Low and the Legendrian Low conjectures. We also show that similar statements hold for any globally hyperbolic $(X^{m+1}, g)$ such that a cover of its Cauchy surface is diffeomorphic to an open domain in $\mathbb R^m.$

Rulings of Legendrian knots as spanning surfaces

Each ruling of a Legendrian link can be naturally treated as a surface. Forknots, the ruling is 2-graded if and only if the surface is orientable. For2-graded rulings of homogeneous (in particular, alternating and positive)knots, we show that the genus of this surface is at most the genus of theknot. While this is not true in general, we do prove that the canonical genusof any knot is an upper bound for the genera of its 2-graded rulings.

On Legendrian Surgeries

We use the Ozsv\'ath-Szab\'o contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact $3$-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskorn homology spheres $-\Sigma(2,3,6n-1)$.

Legendrian links and the spanning tree model for Khovanov homology

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.

Generating family invariants for Legendrian links of unknots

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 polynomials allow one to show that some topologically equivalent links with the same classical invariants are not Legendrian equivalent. It is also shown that for these families of links the generating family polynomials agree with the polynomials arising from a linearization of the differential graded algebra associated to the links.

Links in an open book decomposition and in the standard contact structure

We study a relationship between arc presentations of links in R 3 \mathbb {R}^3 and Legendrian links in R 3 \mathbb {R}^3 with the standard tight contact structure. We determine the arc indices of torus knots.

The Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: The Fuchs conjecture and beyond

Author(s): Rutherford, Dan | Abstract: 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.

A Legendrian Thurston–Bennequin bound from Khovanov homology

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.

Contact homology and one parameter families of Legendrian knots

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) of smooth knots. For this result, we also compute the contact homology of what we call the Legendrian closure of a positive braid and construct an augmentation for each such link diagram.

A Legendrian surgery presentation of contact 3-manifolds

We prove that every closed, connected contact 3-manifold can be obtained from the 3-sphere with its standard contact structure by contact surgery of coefficient plus or minus 1 along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in slightly stronger form).

Legendrian solid-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 al- gebras can be associated to such links. The theory is applied to dis- tinguish 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 horizontal flypes that have the same polynomials but distinct characteristic algebras. Results ob- tainable from a Legendrian satellite construction are compared to results obtainable from the DGA and generating function techniques.

Differential algebra of Legendrian links

Let the space R = {(q, p, u)} be equipped with the standard contact form α = du − pdq. A link L ⊂ R3 is called Legendrian if the restriction of α to L vanishes. Two Legendrian links are said to be Legendrian isotopic if they can be connected by a path in the space of Legendrian embeddings. Consider the map π : R3 → R2, (q, p, u) → (q, p). Since the kernel of dπ is everywhere transverse to the kernel of α, the projection π(L) of a Legendrian link L is an immersed curve. We say that a link L ⊂ R3 is generic with respect toπ, or π-generic, if all self-intersections of π(L) are transverse double points. Given a π-generic Legendrian link L , its diagram is the curve π(L) ⊂ R2, at every crossing of which the overpassing branch (the one with the larger value of u) is marked. It is well known that every smooth link type can be realized by a Legendrian link (it is shown in Sect. 11 that such a realization can be chosen to have a diagram with relatively few crossings). The so-called classical invariants of an oriented Legendrian link L are defined as follows. First of them is just the smooth isotopy type of L . Let L1, . . . , Lk be the components of L . The Maslov number m(Li) of Li is twice the rotation number of π(Li). The Thurston–Bennequin number β(Li) of Li is the linking number between Li and s(Li), where s is a small shift along the u direction. The orientation on R3 involved in the definition of the linking number is given by the form α ∧ dα = dq ∧ dp ∧ du. In terms of the diagram, β(Li) is the number of double points of π(Li) counted with signs shown in Fig. 1. The Thurston–Bennequin number does not depend on the orientation of Li , while the Maslov number changes its sign when the orientation changes.

Generating function polynomials for legendrian links

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 support the belief that these polynomials carry the same information as a refined version of Chekanov's first order polynomials which are defined via the theory of holomorphic curves.

The <b>N</b>-copy of a topologically trivial Legendrian knot

We consider Legendrian knots and links in the standard 3-dimensional contact space. In 1997 Chekanov [Ch] introduced a new invariant for these knots. At the same time, a similar construction was suggested by Eliashberg [E1] within the framework of his joing work with Hofer and Givernthal on Symplectic Field Theory ([E2],[EGH]). To a knot diagram, they associated a differential algebra A. Its stable isomorphism type is invariant under Legendrian isotopy of the knot. In this paper, we introduce an additional structure on this algebra in the case of a Legendrian link. For a link of N components, we show that its algebra splits A = ⊕g ∈ G Ag Here G is a free group on (N - 1) variables. The splitting is determined by the order of the knots and is preserved by the differential. It gives a tool to show that some permutations of link components are impossible to produce by Legendrian isotopy.

A note on contact structures

We construct two contact structures on a homology sphere M , that are homotopic through 2-plane fields, but are not isotopic as contact structures. A diffeomorphism of M permutes these two contact structures. In [LM] Lisca and Matic gave examples of homotopic but not isotopic contact structures. In this note we give a simple construction of another such example. Unlike the examples of Lisca and Matic, in our example the group of diffeomorphism of the 3-manifold acts nontrivially on the set of path components of oriented contact structures of the manifold, while the action of this group is trivial on the set of path components of the oriented tangent plane distributions. Our construction of contact structures uses theorem of Y. Eliashberg, characterizing symplectic 4-manifolds with pseudo-convex boundary (PC -manifolds). Theorem 1 ([E]) : Let X = B ∪ (1− handles) ∪ (2− handles) be fourdimensional handlebody with one 0-handle and no 3or 4-handles. Then: • The standard PC structure on B can be extended over 1-handles so that manifold X1 = B 4 ∪ (1− handles) has pseudo-convex boundary. • If each 2-handle is attached to ∂X1 along a Legendrian knot with framing one less then Thurston-Bennequin framing of this knot, then the symplectic form on X1 can be extended over 2-handles to a symplectic form on X, which makes X a PC manifold. We can visualize pseudo-convex handlebodies as follows: Take a 0-handle to be the unit ball B in C with the induced symplectic structure; choose coordinates in R ⊂ S = ∂B, so that the induced contact structure ξ0 on R is the kernel of the form λ0 = dz+ xdy. We also assume that the centers of the attaching balls of each 1-handle lies on the plane {x = constant}. Projection of a Legendrian link on the plane {x = 0} has intersection only of the type shown on Figure 1 (coming from left handed crossing), no vertical tangencies and all minima and maxima in y-direction are cusps. Moreover, every projection with these properties is a projection of some Legendrian link. Thurston-Bennequin invariant of a Legendrian knot α can be calculated by: tb(α) = bb(α)− (number of right cusps) where bb(α) is the blackboard (yz-plane) framing of the projection of α. Any PC-manifoldX induces a contact structure on the boundary 3-manifold Y = ∂X by restricting the dual K∗ of the canonical line bundleK → X. Furthermore if α

Legendrian circular helix links

Examples are given of legendrian links in the manifold of cooriented contact elements of the plane, or equivalently, in the 1-jet space of the circle which are not equivalent via an isotopy of contact diffeomorphisms. These examples have generalizations to linked legendrian spheres in contact manifolds diffeomorphic to ℝn×Sn−1. These links are distinguished by applying the theory of generating functions to contact manifolds.

Estimates for the Bennequin number of Legendrian links from state models for knot polynomials

A contact structure in a 3-dimensional manifold is a completely nonintegrable 2-dimensional distribution: if the distribution is the kernel of a (locally defined) 1-form λ then λ∧dλ � 0everywhere. The standard contact structure in 3-space, arising from the identification of R 3 with the space of 1-jets of functions on the line, is given by the contact form λ = dz − yd x; here x is a point on the line, z is the value of a function and y the value of the derivative at x. The same formula defines the standard contact structure in the space of 1-jets of functions on the circle (x being the cyclic coordinate), the space which is topologically the solid torus. In this paper we will be concerned only with these two contact manifolds. In a contact 3-dimensional manifold one considers two classes of knots and links: the transverse and the Legendrian ones. The former are everywhere transverse to the contact distribution, and the latter are everywhere tangent to it. Every topological isotopy class of knots in a contact 3-fold contains a transverse and a Legendrian knot. The main problem of contact knot theory (which has made, so far, only a few first steps) is to classify, up to contact isotopies, transverse and Legendrian knots within each topological isotopy class. In particular, one would like to have specifically contact invariants of transverse and Legendrian knots and links. One such invariant is easily defined. Legendrian and transverse knots in the standard contact space have natural framings given by the vector fields ∂z and ∂y, respectively. The corresponding self-linking number is called the Bennequin number; it is denoted by β(K) where K is a Legendrian or a transverse knot 1 . Similarly one defines the Bennequin number of an oriented Legendrian or transverse link. The study of knots in contact 3-dimensional manifolds was put forward by the seminal paper by D. Bennequin [Be]. One of the main results of this paper