Thurston-Bennequin Number Research Articles

Distinguishing Legendrian knots in topological types 74, 948, 10136 with maximal Thurston–Bennequin number

In a recent work of Dynnikov and Prasolov [I. Dynnikov and M. Prasolov, An algorithm for comparing Legendrian knots, preprint (2023), arXiv:2309.05087] a new method of comparing Legendrian knots with nontrivial symmetry group is proposed. Using this method, we confirm conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997] about Legendrian knots in topological types [Formula: see text], [Formula: see text] and [Formula: see text]. This completes the classification of Legendrian types of rectangular diagrams of knots of complexity up to 9.

Bi-Legendrian rack colorings of Legendrian knots

We introduce a bi-Legendrian rack and show that a bi-Legendrian rack coloring number is an invariant of Legendrian knots. We prove that bi-Legendrian rack coloring numbers can distinguish all Legendrian unknots with the same Thurston–Bennequin number. We also consider pairs of Legendrian knots which cannot be distinguished by bi-Legendrian rack coloring numbers.

Crossing numbers and rotation numbers of cycles in a plane immersed graph

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 [Formula: see text]-cycles is odd. The sum of the rotation numbers of all [Formula: see text]-cycles is even. We show analogous results for [Formula: see text]-cycles, [Formula: see text]-cycles and [Formula: see text]-cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a [Formula: see text]-cycle that is not an unknot with maximal Thurston–Bennequin number, and the sum of all Thurston–Bennequin numbers of the cycles is seven times the sum of all Thurston–Bennequin numbers of the [Formula: see text]-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that have a generic immersion into a plane whose all cycles have rotation number [Formula: see text].

Genus One Cobordisms Between Torus Knots

Abstract We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $\nu ^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston–Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.

Classification of Legendrian knots of topological type 76 with maximal Thurston-Bennequin number

Classification of Legendrian knots of topological type 7₆ with maximal Thurston-Bennequin number

Extremal Khovanov homology of Turaev genus one links

The Turaev genus of a link can be thought of as a way of measuring how non-alternating a link is. A link is Turaev genus zero if and only if it is alternating, and in this viewpoint, links with large Turaev genus are very non-alternating. In this paper, we study Turaev genus one links, a class of links which includes almost alternating links. We prove that the Khovanov homology of a Turaev genus one link is isomorphic to $\mathbb{Z}$ in at least one of its extremal quantum gradings. As an application, we compute or nearly compute the maximal Thurston Bennequin number of a Turaev genus one link.

Classification of Legendrian knots of topological type 76 with maximal Thurston–Bennequin number

We classify Legendrian knots of topological type [Formula: see text] having maximal Thurston–Bennequin number confirming the corresponding conjectures of [W. Chongchitmate and L. Ng, An atlas of Legendrian knots, Exp. Math. 22(1) (2013) 26–37, arXiv:1010.3997].

Concordance invariants of doubled knots and blowing up

Let $\nu$ be either the Ozsvath-Szabo $\tau$-invariant or the Rasmussen $s$-invariant, suitably normalized. For a knot $K$, Livingston and Naik defined the invariant $t_\nu(K)$ to be the minimum of $k$ for which $\nu$ of the $k$-twisted positive Whitehead double of $K$ vanishes. They proved that $t_\nu(K)$ is bounded above by $-TB(-K)$, where $TB$ is the maximal Thurston-Bennequin number. We use a blowing up process to find a crossing change formula and a new upper bound for $t_\nu$ in terms of the unknotting number. As an application, we present infinitely many knots $K$ such that the difference between Livingston-Naik's upper bound $-TB(-K)$ and $t_\nu(K)$ can be arbitrarily large.

Nonfillable Legendrian knots in the 3–sphere

If a Legendrian knot $\Lambda$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $\Sigma$ in the standard symplectic 4-ball, then the genus of $\Sigma$ is equal to the slice genus of (the smooth knot underlying) $\Lambda$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $\Sigma$ is zero as well as the rotation number of $\Lambda$, and moreover, the linearized contact homology of $\Lambda$ with respect to the augmentation induced by $\Sigma$ is isomorphic to the (singular) homology of $\Sigma$. It was asked in arXiv:1212.1519 whether the converse of this statement is true. We give a negative answer to this question by providing a family of Legendrian knots with augmentations not induced by any exact Lagrangian filling although the associated linearized contact homology is isomorphic to the homology of the smooth surface of minimal genus in the 4-ball bounding the knot.

Morse structures on open books

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 Legendrian knot from its front projection.

Exact Lagrangian fillings of Legendrian (2,n) torus links

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and K\'alm\'an 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.

Rectangular diagrams of surfaces: representability

Introduced here is a simple combinatorial way, which is called a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on and to rectangular diagrams of links. By using rectangular diagrams of surfaces it is intended, in particular, to develop a method to distinguish Legendrian knots. This requires a lot of technical work of which the present paper addresses only the first basic question: which isotopy classes of surfaces can be represented by a rectangular diagram? Roughly speaking, the answer is this: there is no restriction on the isotopy class of the surface, but there is a restriction on the rectangular diagram of the boundary link arising from the presentation of the surface. The result extends to Giroux's convex surfaces for which this restriction on the boundary has a natural meaning. In a subsequent paper, transformations of rectangular diagrams of surfaces will be considered and their properties will be studied. By using the formalism of rectangular diagrams of surfaces an annulus in is produced here that is expected to be a counterexample to the following conjecture: if two Legendrian knots cobound an annulus and have zero Thurston-Bennequin numbers relative to this annulus, then they are Legendrian isotopic. Bibliography: 30 titles.

Nonexistence of Stein structures on 4-manifolds and maximal Thurston–Bennequin numbers

For a 4-manifold represented by a framed knot in $S^3$, it has been well known that the 4-manifold admits a Stein structure if the framing is less than the maximal Thurston-Bennequin number of the knot. In this paper, we prove either the converse of this fact is false or there exists a compact contractible oriented smooth 4-manifold (with Stein fillable boundary) admitting no Stein structure. Note that an exotic smooth structure on $S^4$ exists if and only if there exists a compact contractible oriented smooth 4-manifold with $S^3$ boundary admitting no Stein structure.

Geometry of Legendrian knots

We study the extrinsic geometry of Legendrian knots in the standard tight contact structure on [Formula: see text] In particular, we show that the total curvature of a Legendrian knot [Formula: see text] in [Formula: see text] is bounded below by [Formula: see text] times, the total number of cusps in the front projection of [Formula: see text]. We also show that a Legendrian [Formula: see text]-torus knot has the total curvature bounded below by [Formula: see text] while that of the Legendrian knots [Formula: see text] is bounded below by [Formula: see text]. Furthermore, we find an explicit relation between the Thurston–Bennequin number of a Legendrian knot [Formula: see text] and the geometric self-linking number, the curvature and the torsion of the knot [Formula: see text].

Maximal Thurston–Bennequin number and reducible Legendrian surgery

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.

Contact structures and reducible surgeries

We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus$g$must have slope$2g-1$, leading to a proof of the cabling conjecture for positive knots of genus 2. Our techniques also produce bounds on the maximum Thurston–Bennequin numbers of cables.

Lagrangian concordance is not a symmetric relation

We provide an explicit example of a non trivial Legendrian knot \Lambda such that there exists a Lagrangian concordance from \Lambda_0 to \Lambda where \Lambda_0 is the trivial Legendrian knot with maximal Thurston–Bennequin number. We then use the map induced in Legendrian contact homology by a concordance and the augmentation category of \Lambda to show that no Lagrangian concordance exists in the other direction. This proves that the relation of Lagrangian concordance is not symmetric.

On the maximal Thurston–Bennequin number of knots and links in spatial graphs

It is known that there does not exist a Legendrian embedding of K4 such that all its cycles realize the maximal Thurston–Bennequin numbers that only consist of odd numbers. This paper represents an infinite family of spatial graphs that are Legendrian embeddings of K4 such that, for each of the Legendrian embeddings, all its cycles but one realize the maximal Thurston–Bennequin numbers that consist of odd numbers and its one cycle realizes the maximal Thurston–Bennequin number equal to zero.

Legendrian𝜃-graphs

In this article we give necessary and sufficient conditions for two triples of integers to be realized as the Thurston-Bennequin number and the rotation number of a Legendrian theta-graph with all cycles unknotted. We show that these invariants are not enough to determine the Legendrian class of a topologically planar theta-graph. We define the transverse push-off of a Legendrian graph and we determine its self linking number for Legendrian theta-graphs. In the case of topologically planar theta-graphs, we prove that the topological type of the transverse push-off is that of a pretzel link.

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.