arXiv research

The study estimates Reeb chords using sheaf theory and persistence.

In this article, we prove that there exists at least one chord which is characteristic of Reeb vector field connecting a given Legendre submanifold in a closed contact manifold with any contact form.

Assume that we are given a closed chord-generic Legendrian submanifold $Λ\subset P \times \mathbb R$ of the contactisation of a Liouville manifold, where $Λ$ moreover admits an exact Lagrangian filling $L_Λ \subset \mathbb R \times P \times \mathbb R$ inside the symplectisation. Under the further assumptions that this …

We prove that the number of Reeb chords between a Legendrian submanifold and its contact Hamiltonian push-off is at least the sum of the $\mathbb{Z}_2$-Betti numbers of the submanifold, provided that the contact isotopy is sufficiently small when compared to the smallest Reeb chord on the Legendrian. Moreover, the esta…

Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.

New algebra invariant distinguishes Legendrian knots in convex surfaces.

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geode…

New techniques reveal tight contact manifolds with vanishing contact homology.

The paper calculates a formula for knot complements using holomorphic curves.

One way to obtain invariants of some Legendrian submanifolds in 1-jet spaces $J^1M$, equipped with the standard contact structure, is through the Morse theoretic technique of generating families. This paper extends the invariant of generating family cohomology by giving it a product $μ_2$. To define the product, moduli…

The paper shows that knot projections without triple chords can be simplified.

Paper explores relationships between triple chords and a specific homotopy relation in knot theory.

Solves $L_p$-Gaussian chord Minkowski problem using Gauss curvature flow.

The abstract introduces a new $A_\infty$ duality via LSFT algebra.

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary cir…

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define…

A chord index homomorphism for knots in thickened surfaces is constructed.

The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.

Paper solves a generalized chord Minkowski problem using Gauss curvature flows.

In this paper we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to define finite type invariants of virtual knots. The notions of indexed Jones polynomial and indexed quandle are introduced, which generalize the classical Jones pol…

Study contact instantons and Legendrian links, proving energy inequalities.

Enhances Vassiliev knot invariants using chord diagrams.

Study counts sub-chord diagrams to classify spherical curves.

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus o…

In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, th…

We propose a novel approach for the generation of polyphonic music based on LSTMs. We generate music in two steps. First, a chord LSTM predicts a chord progression based on a chord embedding. A second LSTM then generates polyphonic music from the predicted chord progression. The generated music sounds pleasing and harm…

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spann…

Paper develops invariants for spherical curves using chord diagrams.

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loo…

The paper calculates a specific weight system for chord diagrams with a particular graph structure.

We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and…

The paper finds an upper limit for the length of geodesic chords on Riemannian manifolds.

We present a new approach to harmonic analysis that is trained to segment music into a sequence of chord spans tagged with chord labels. Formulated as a semi-Markov Conditional Random Field (semi-CRF), this joint segmentation and labeling approach enables the use of a rich set of segment-level features, such as segment…

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to t…

Proves Arnol'd's chord conjecture for conormal bundles.

Invariant detects sliceness of virtual knots with specific chord indices.

The study proves geodesic loops and chords without intersections for specific metrics.

Study on curve shortening flow with boundary conditions, proving convergence or contraction.

The paper defines functions from spherical curves and chord diagrams, proving invariance under specific Reidemeister moves.

Complete invariant defined for doodles on a sphere.

In previous work, we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term rela…

Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.

The paper defines new homotopy relations on knot projections and classifies certain knot types.

Optimal Reeb graphs identified for polygon decomposition.

Chord diagrams on circles and their intersection graphs (also known as circle graphs) have been intensively studied, and have many applications to the study of knots and knot invariants, among others. However, chord diagrams on more general graphs have not been studied, and are potentially equally valuable in the study…

Characterizes smooth functions on manifolds with simple Reeb spaces.