Legendrian Knots Research Articles

Poincaré series and linking of Legendrian knots

Poincaré series and linking of Legendrian knots

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.

Non-Orientable Lagrangian Fillings of Legendrian Knots

AbstractWe investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

Conjectures about virtual Legendrian knots and links

We formulate conjectures generalizing some known results to the category of virtual Legendrian knots. This includes statements relating virtual Legendrian knots to ordinary Legendrian knots, non-existence of positive virtual Legendrian self isotopies for the class of the fiber of ST⁎M and the conjectural relation of virtual Legendrian isotopies to causality in generalized spacetimes. We prove the conjectures in the case of 2-dimensional M and (2+1)-dimensional spacetimes. We also formulate and prove the version of the Arnold's 4 cusp conjecture for virtual isotopies.

Distinguishing Legendrian knots with trivial orientation-preserving symmetry group

Distinguishing Legendrian knots with trivial orientation-preserving symmetry group

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.

On Cusps of Caustics by Reflection: Billiard Variations on the Four Vertex Theorem and on Jacobi’s Last Geometric Statement

A point source of light is placed inside an oval. The nth caustic by reflection is the envelope of the light rays emanating from the light source after n reflections off the curve. We show that, for a generic point light source, each of these caustics has at least 4 cusps. This is a billiard variation on Jacobi’s Last Geometric Statement concerning the number of cusps of the conjugate locus of a point on a convex surface. We present various proofs, using different ideas, including the curve shortening flow and Legendrian knot theory.

On a generalization of Jones polynomial and its categorification for Legendrian knots

In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade shift.The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.

Legendrian satellites and decomposable cobordisms

In this paper, we investigate the interactions between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots. Given Lagrangian cobordisms between two Legendrian knots and between two Legendrian tangles, we construct a Lagrangian cobordism between Legendrian satellites of the knots by the closures of the tangles, with extra twists on both the top and the bottom satellite to compensate for the genus of the cobordism. If the original cobordisms were decomposable, then a decomposable cobordism between satellites exists as well, again with extra twists.

A combinatorial description of the LOSS Legendrian knot invariant

In this paper, we observe that the hat version of the Heegaard Floer invariant of Legendrian knots in contact three-manifolds defined by Lisca-Ozsváth-Stipsicz-Szabó can be combinatorially computed. We rely on Plamenevskaya’s combinatorial description of the Heegaard Floer contact invariant.

Planar Legendrian Θ ${\rm{\Theta }}$‐graphs

AbstractWe classify topologically trivial Legendrian ‐graphs in and identify the complete family of nondestabilizable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of a ‐graph or as a subgraph will have an infinite number of distinct, topologically trivial nondestabilizable Legendrian embeddings. Additionally, we introduce two moves, vertex stabilization and vertex twist, that change the Legendrian type within a smooth isotopy class.

On Contact Surgery and Knot Floer Invariants

Abstract We establish some general relations between Heegaard-Floer-based contact invariants. In particular, we observe that if the contact invariant of large negative, respectively positive, contact surgeries along a Legendrian knot does not vanish, then the Legendrian invariant, respectively the Legendrian inverse limit invariant, of that knot is non-zero. We use sutured Floer homology and the limit constructions due to Golla and Etnyre, Vela-Vick and Zarev.

Concordance maps in HFK−

We show that a decorated knot concordance [Formula: see text] from [Formula: see text] to [Formula: see text] induces an [Formula: see text]-module homomorphism [Formula: see text] which preserves the Alexander and absolute [Formula: see text]-Maslov gradings. Our construction generalizes the concordance maps induced on [Formula: see text] studied by Juhász and Marengon [Concordance maps in knot Floer homology, Geom. Topol. 20 (2016) 3623–3673], but uses the description of [Formula: see text] as a direct limit of maps between sutured Floer homology groups discovered by Etnyre et al. [Sutured Floer homology and invariants of Legendrian and transverse knots, Geom. Topol. 21 (2017) 1469–1582].

New invariants of Legendrian knots

We give new functions of Legendrian knots derived from Legendrian fronts. These are integer-valued linear functions that are alike the Arnold basic invariant of plane curves. Various generalizations of the Arnold basic invariant have been known. In this paper, we give another extension of Arnold's idea.

ON ELEMENTARY MOVES OF SINGULAR LEGENDRIAN KNOTS

We have two results. First, we give 96 generating sets oriented singular Reidemeister moves; it is an answer to a question by Bataineh, Khaled, Elhamdadi, and Hajij who give a generating set of oriented singular Reidemeister moves using their computation. Second, in the theory of plane curve and Legendrian knots introduced by V. I. Arnold, we select which moves survive as those of Legendrian singular knots and fronts diagrammatically and explicitly.

Geography of Legendrian knot mosaics

In this paper, mosaic tiles have been used to build knots and define invariants since they were introduced by Lomonaco and Kauffman to describe quantum knots. In this paper, we propose a modified set of tiles to describe the front projections of Legendrian knots. We explore the effect of stabilization on the mosaic number of a Legendrian knot by defining and classifying “two-cell” stabilizations. Inspired by a construction of Ludwig et al. in 2013, we provide an infinite family of Legendrian unknots whose mosaic numbers are realized only in non-reduced projections. Finally, we provide a census of known mosaic numbers for Legendrian unknots and trefoils.

On long knots in the full torus

The aim of this paper is to realize the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of particular interest because of their relation to Legendrian knots, knotoids, [Formula: see text]-manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [V. O. Manturov, A free-group valued invariant of free knots, preprint (2020), arXiv:2012.15571v2]. Long knots naturally appear in the study of classical knots [T. Fiedler, More [Formula: see text]-cocycles for classical knots, preprint (2020), arXiv:2004.04624; A. Mortier, A Kontsevich integral of order [Formula: see text], preprint (2018), arXiv:1810.05747].

Non-fillable Augmentations of Twist Knots

Abstract We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel –Ekholm–Dimitroglou Rizell isomorphism with local coefficients to show that any Lagrangian filling point in the augmentation variety of a Legendrian knot must lie in the injective image of an algebraic torus with dimension equal to the 1st Betti number of the filling. This is a Floer-theoretic version of a result from microlocal sheaf theory. For the augmentations in question, we show that no such algebraic torus can exist.

Sheaves via augmentations of Legendrian surfaces

Given an augmentation for a Legendrian surface in a 1-jet space, $$\Lambda \subset J^1(M)$$ , we explicitly construct an object, $$\mathcal {F} \in \mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})$$ , of the (derived) category from Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) of constructible sheaves on $$M\times \mathbb {R}$$ with singular support determined by $$\Lambda $$ . In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in 1-jet spaces that, based on Rutherford and Sullivan (Cellular Legendrian contact homology for surfaces, Part I, arXiv:1608.02984 .) Rutherford and Sullivan (Internat J Math 30(7):135, 2019) Rutherford and Sullivan (Internat J Math 30(7):111, 2019), is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of Shende, Treumann and Zaslow (Invent Math 207(3), 1031–1133 (2017)) for 1-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $$\mathbf {Sh}^\bullet _{\Lambda }(M\times \mathbb {R}, \mathbb {K})$$ in the 2-dimensional case.

Rectangular diagrams of surfaces: distinguishing Legendrian knots

In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce moves that allow transition between diagrams representing isotopic surfaces. We also introduce more general combinatorial objects called mirror diagrams and various moves for them that can be used to transform presentations of isotopic surfaces to each other. The moves can be divided into two types so that, vaguely speaking, type~I moves commute with type~II ones. This commutation is the matter of the main technical result of the paper. We use it as well as a relation of the moves to Giroux's convex surfaces to propose a new method for distinguishing Legendrian knots. We apply this method to show that two Legendrian knots having topological type $6_2$ are not equivalent. More applications of the method will be a subject of subsequent papers.