Legendrian Knots Research Articles

Ruling polynomials and augmentations over finite fields

For any Legendrian link, L, in ( R 3 , ker ( d z - y d x ) ) , we define invariants, Aug m ( L , q ) , as normalized counts of augmentations from the Legendrian contact homology differential graded algebra (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 from [Ng and Sabloff, ‘The correspondence between augmentations and rulings for Legendrian knots’, Pacific J. Math. 224 (2006) 141–150] for the case q = 2 , we show the augmentation numbers, Aug m ( L , q ) , are determined by specializing the m-graded ruling polynomial, R L m ( 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.

Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions

In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. It is shown that a minimal rectangular diagram maximizes the Thurston-Bennequin number for the corresponding Legendrian links. Jones' conjecture about the invariance of the algebraic number of intersections of a minimal braid representing a fixed link type is proved. A new proof of the monotonic simplification theorem for the unknot is given.

The surgery unknotting number of Legendrian links

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 Ng whose underlying smooth knot has crossing number 7 or less. In all these calculations, as long as the Legendrian link of $j$ components is not topologically a slice knot, its surgery unknotting number is equal to the sum of $(j-1)$ and twice the smooth 4-ball genus of the underlying smooth link.

Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra

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 type and Thurston-Bennequin number of the knot.

Loose Legendrians and the plastikstufe

We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n+1>3. More precisely, we prove that every Legendrian knot whose complement contains a "nice" plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two non-isomorphic contact structures become isomorphic after connect-summing with a manifold containing a plastikstufe.

On knots in overtwisted contact structures

We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show similar results for Legendrian knots and prove a “folk” result concerning loose transverse and Legendrian knots (that is knots with overtwisted complements) which says that such knots are determined by their classical invariants (up to contactomorphism). Finally we discuss how these results partially fill in our understanding of the “geography” and “botany” problems for Legendrian knots in overtwisted contact structures, as well as many open questions regarding these problems.

Legendrian contact homology in Seifert fibered spaces

We define a differential graded algebra associated to a Legendrian knot in a Seifert fibered space with a transverse contact structure. The new feature of this construction is the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as in the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology is torsion and whose underlying topological knot types are isotopic; such examples exist in any sufficiently complicated contact Seifert fibered space.

A COMBINATORIAL DGA FOR LEGENDRIAN KNOTS FROM GENERATING FAMILIES

For a Legendrian knot L ⊂ ℝ3, with a chosen Morse complex sequence (MCS), we construct a differential graded algebra (DGA) whose differential counts "chord paths" in the front projection of L. The definition of the DGA is motivated by considering Morse-theoretic data from generating families. In particular, when the MCS arises from a generating family F, we give a geometric interpretation of our chord paths as certain broken gradient trajectories which we call "gradient staircases". Given two equivalent MCS's we prove the corresponding linearized complexes of the DGA are isomorphic. If the MCS has a standard form, then we show that our DGA agrees with the Chekanov–Eliashberg DGA after changing coordinates by an augmentation.

The contact homology of Legendrian knots with maximal Thurston–Bennequin invariant

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.

THE SUPPORT GENERA OF CERTAIN LEGENDRIAN KNOTS

In this paper, the support genera of all Legendrian right-handed trefoil knots and some other Legendrian knots are computed. We give examples of Legendrian knots in the three-sphere with the standard contact structure which have positive support genera with arbitrarily negative Thurston–Bennequin invariant. This answers a question in [S. Onaran, Invariants of Legendrian knots from open book decompositions, Int. Math. Res. Not.10 (2010) 1831–1859].

Legendrian and transverse cables of positive torus knots

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" tori, which allow us to completely classify solid tori representing positive torus knots.

Rational Seifert surfaces in Seifert fibered spaces

Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we generalize Seifert's algorithm to explicitly construct a rational Seifert surface for any rationally null-homologous knot. As an application of the techniques developed in the paper, we derive closed formulae for the rational Thurston-Bennequin and rotation numbers of a Legendrian knot in a contact Seifert fibered space.

Monopole Floer homology and Legendrian knots

We use monopole Floer homology for sutured manifolds to construct invariants of Legendrian knots in a contact 3-manifold. These invariants assign to a knot K in Y elements of the monopole knot homology KHM(-Y,K), and they strongly resemble the knot Floer homology invariants of Lisca, Ozsv\'ath, Stipsicz, and Szab\'o. We prove several vanishing results, investigate their behavior under contact surgeries, and use this to construct many examples of non-loose knots in overtwisted 3-manifolds. We also show that these invariants are functorial with respect to Lagrangian concordance.

CUBE NUMBER CAN DETECT CHIRALITY AND LEGENDRIAN TYPE OF KNOTS

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. Examples of knots for which the cube number detects chirality are presented. An example is also presented of two Legendrian knots that, while topologically isotopic, are distinguished as Legendrian knots by cube number.

Neighbourhoods and isotopies of knots in contact 3-manifolds

We prove a neighbourhood theorem for arbitrary knots in contact 3-manifolds. As an application we show that two topologically isotopic Legendrian knots in a contact 3-manifold become Legendrian isotopic after suitable stabilisations.

On the Transverse Invariant for Bindings of Open Books

Let $T \subset (Y,\xi)$ be a transverse knot which is the binding of some open book, $(T,\pi )$, for the ambient contact manifold $(Y,\xi)$. In this paper, we show that the transverse invariant $\hat{\mathscr{T}(T) \in \widehat{HFK}(−Y,K)$, defined in P. Lisca, P. Ozsvath, A. I. Stipsicz & Z. Szabo, Heegaard Floer invariants of Legendrian knots in contact three-manifolds, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 6, 1307–1363, MR 2557137, Zbl pre05641376, is nonvanishing for such transverse knots. This is true regardless of whether or not $\xi$ is tight. We also prove a vanishing theorem for the invariants $\mathscr{L}$ and $\mathscr{T}$. As a corollary, we show that if $(T,\pi )$ is an open book with connected binding, then the complement of $T$ has no Giroux torsion.

INVARIANTS FOR LEGENDRIAN KNOTS IN LENS SPACES

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra [Formula: see text] which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S1-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

We define an algebraic/combinatorial object on the front projection $\Sigma$ of a Legendrian knot called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on $\Sigma$ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of $L_\Sigma$, where $L_\Sigma$ is the Ng resolution of $\Sigma$. In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms.

Connections between Floer-type invariants and Morse-type invariants of Legendrian knots

We define an algebraic/combinatorial object on the front projection Σ of a Legendrian knot, called a Morse complex sequence, abbreviated MCS. This object is motivated by the theory of generating families and provides new connections between generating families, normal rulings, and augmentations of the Chekanov-Eliashberg DGA. In particular, we place an equivalence relation on the set of MCSs on Σ and construct a surjective map from the equivalence classes to the set of chain homotopy classes of augmentations of L Σ , where L Σ is the Ng resolution of Σ. In the case of Legendrian knot classes admitting representatives with two-bridge front projections, this map is bijective. We also exhibit two standard forms for MCSs and give explicit algorithms for finding these forms. The definition of an MCS, the equivalence relation, and the statements of some of the results originate from unpublished work of Petya Pushkar.

A bordered Chekanov-Eliashberg algebra

Given a front projection of a Legendrian knot $K$ in $\mathbb{R}^{3}$ which has been cut into several pieces along vertical lines, we assign a differential graded algebra to each piece and prove a van Kampen theorem describing the Chekanov-Eliashberg invariant of $K$ as a pushout of these algebras. We then use this theorem to construct maps between the invariants of Legendrian knots related by certain tangle replacements, and to describe the linearized contact homology of Legendrian Whitehead doubles. Other consequences include a Mayer-Vietoris sequence for linearized contact homology and a van Kampen theorem for the characteristic algebra of a Legendrian knot.