We introduce a new cohomology-theoretic method for classifying generic immersed curves in closed compact surfaces by using Gauss codes. This subsumes a result of J.S. Carter on classifying immersed curves in oriented compact surfaces, and provides a criterion for when an immersion is two-colorable. We note an applicati…
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48 results for “immersed curve theory”
Study on singularities of Lagrangian immersions with applications in Floer theory.
problem Understanding singularities of Lagrangian immersions.
method Applying Hamiltonian isotopy in the Weinstein tubular neighbourhood to express singular points as fold points with cusp points.
result Local expression of singular points of Lagrangian immersions as fold points with cusp points.
The paper proves existence of minimal homotopies for immersed planar curves.
problem Existence of area-minimizing homotopies between homotopic curves in the plane.
method Geometric and variational approach, lifting curves into higher co-dimension, applying Douglas's solution of the Plateau problem.
result Uniform convergence of Douglas minimizers and minimal homotopy area minimization.
Geometric interpretation of tangle invariants using immersed curves.
problem Understanding tangle invariants in Khovanov homology.
method Geometric interpretation of Bar-Natan's invariant using immersed curves.
result Recovery of Khovanov homology from immersed curves.
Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
Paper proves existence of isometric immersions for negatively curved surfaces with unbounded second fundamental form.
problem Existence of isometric immersions for surfaces with negative Gaussian curvature.
method Reformulated Gauss--Codazzi equations into hyperbolic conservation laws, applied theories of invariant regions and compensated compactness.
result Established existence of -isometric immersions for various families of metrics.
The paper tackles deeply slice knots via immersed curves.
problem Whether there exists a homology 3-sphere with certain knot properties.
method Constructing knots in contractible 4-manifolds and using Heegaard Floer homology invariants.
result Proves the existence of a class of deeply slice knots.
A few years ago N.A'Campo invented a construction of a link from a real curve immersed into a disk. In the case of the curve originating from the real morsification method the link is isotopic to the link of the corresponding singularity. There are some curves which do not occur in the singularity theory. In this artic…
Proves a conjecture about concordance invariant simplifying its relation to Rasmussen's invariant.
problem Concordance invariant and its relation to Rasmussen's invariant.
method Bar-Natan's tangle version of Khovanov homology, distilled into immersed curve theory.
result Simplifies the relation between and Rasmussen's -invariant.
Curves with constant torsion can be deformed arbitrarily.
problem Deforming curves of constant torsion in Euclidean space.
method Convex integration and degree theory.
result Existence of knots with constant torsion in each isotopy class.
This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-ord…
New knots not slice in rational 4-balls found.
problem Identifying knots not slice in rational homology 4-balls.
method Using generalized Mazur patterns and immersed Heegaard Floer homology.
result Infinitely many examples of pattern knots P not slice in any rational homology 4-ball.
We construct an algebraic version of Lagrangian Floer homology for immersed curves inside the pillowcase. We first associate to the pillowcase an algebra A. Then to an immersed curve L inside the pillowcase we associate an A infinity module M(L) over A. Then we prove that Lagrangian Floer homology HF(L,L') is isomorphi…
The paper develops theory for holomorphic null curves in SL2(C).
problem Developing theory for holomorphic null curves in SL2(C).
method Establish Runge, Mergelyan, Mittag-Leffler, and Carleman type theorems for holomorphic null immersions.
result Proves every open Riemann surface admits a proper holomorphic null embedding into SL2(C).
Diagrams of divide linksmath.GT
Recently N.A'Campo suggested a construction of a link from a generic immersion of a curve into a 2-disk. It is tightly related to the singularity theory. In this paper, we give a simple procedure to draw a diagram of the link from a picture of the curve.
Geometric interpretation of 3-manifold invariants using immersed curves.
problem Obstructing smooth equivalences between 4-manifolds and surfaces with boundary.
method Relating morphisms between bordered Floer invariants to cobordism maps via immersed curves in the punctured torus.
result Morphisms between immersed curve invariants compute certain cobordism maps.
New homomorphism proven using immersed curves on disks.
problem Existence of homomorphisms and summands in homology groups.
method Approximating involutive Heegaard Floer complexes with immersed curves on the twice punctured disk.
result Existence of a new homomorphism and a summand proven.
Framework for isometric immersions of planar regions from framed curves.
problem Characterizing isometric immersions of planar regions with piecewise smooth boundaries.
method Develops a framework using framed curves and compatibility/regularity conditions.
result Exact dimensional reduction of bending energy to a line integral over the boundary curve.
The quaternionic KP hierarchy is the integrable hierarchy of p.d.e obtained by replacing the complex numbers with the quaternions, mutatis mutandis, in the standard construction of the KP hierarchy equations and solutions; it is equivalent to what is often called the Davey-Stewartson II hierarchy. This article studies …
Study disproves conjecture about metric completion of curve spaces.
problem Completeness properties of spaces of immersed curves with reparametrization-invariant metrics.
method Examined Sobolev-type metrics on real-valued immersed curves, demonstrating multiple distinct limit points.
result Metric completion of spaces of immersed open curves includes multiple distinct limit points, not a single point as previously conjectured.
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
We prove short-time existence of φ-regular solutions to the anisotropic and crystalline curvature flow of immersed planar curves.
Space curves with convex projections evolve smoothly until shrinking to a point.
problem Evolution of space curves with convex projections.
method Space Curve Shortening flow.
result Convex projections remain convex throughout the evolution.
In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formu…
Study proves uniqueness of corrugated negatively curved immersions in differential geometry.
problem Negatively curved immersions in differential geometry.
method Relative entropy method applied to Gauss-Codazzi system.
result Uniqueness of smooth isometric immersions within corrugated class.
Analytic curves have infinite codimension of singular germs.
problem Understanding the codimension of singular tangent curves in analytic distributions.
method Formalizing asymptotic statements about finite jets of tangent curves and applying the h-principle.
result The subspace of singular germs has infinite codimension within smooth curves.
We prove that a closed immersed plane curve with total curvature has entropy at least times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature whose entropy is less than …
The study examines how twisting a knot affects its homology and stability properties.
problem Investigating the impact of twisting a knot on its homology and stability.
method Use bordered Floer homology and immersed curve invariants.
result Total dimension, τ(K_m), and thickness of K_m are linear functions of m for large m.
A new method converts knot Floer homology to immersed curves.
problem Calculating bordered invariants of knot complements.
method Handle attachment to convert twice-punctured disk to once-punctured torus.
result Recovers Lipshitz-Ozsváth-Thurston's bordered invariant result.
Algorithm computes Thurston norm for hyperbolic 3-manifolds.
problem Computing the Thurston norm for hyperbolic 3-manifolds.
method Developed a theory of spun-normal immersed surfaces and implemented an algorithm.
result Computed the unit ball of the Thurston norm for cusped hyperbolic 3-manifolds.
Biminimal immersionsmath.DG
We study biminimal immersions, that is immersions which are critical points of the bienergy for normal variations with fixed energy. We give a geometrical description of the Euler-Lagrange equation associated to biminimal immersions for: i) biminimal curves in a Riemannian manifold, with particular care to the case of …
Study how invariants change under bifurcations of curves.
problem Understanding how invariants of curves change under bifurcations.
method Analyzing , , , and invariants of curves under -bifurcations.
result Invariant changes under bifurcations, preserving its essential properties.
We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex Kähler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic deformations of a non-compact complex curve in a complex manifold, parametrized by a finite …
In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves i…
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises as the zero locus of the determinant of a holomorphic family of Dirac type opera…
The study explores holomorphic Legendrian curves and superminimal surfaces in complex projective and sphere spaces.
problem Characterizing and embedding holomorphic Legendrian curves and superminimal surfaces.
method Runge approximation theorem, bijective correspondence via twistor projection, finite genus analysis.
result Every open Riemann surface embeds into as a complete holomorphic Legendrian curve.
The space of all immersed closed curves of rotation degree 0 in the plane modulo reparametrizations has the same homotopy groups as the circle times the 2-sphere.
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…
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct a ternary invariant of curves immersed in compact surfaces, conside…
Study character varieties of tangles to map immersed curves in the pillowcase.
problem Characterizing holonomy-perturbed traceless SU(2) character varieties.
method Examining marked tangles as endomorphisms in the cobordism category and using holonomy-perturbed traceless character variety functor.
result Endomorphisms of immersed curves in the pillowcase have the same image.
Revisits Weyl's problem on isometric immersions of spheres into 3D manifolds.
problem Isometric immersions of S² into 3D Riemannian manifolds with non-negative Gauss curvature.
method Utilizes the framework of J-holomorphic curves developed by Labourie.
result Exhibits a sufficient condition for the existence of global C¹¹ isometric immersions.
Study corrects previous work on knot Floer homology of certain pretzel knots.
problem Computing the Knot Floer Homology of specific pretzel knots.
method Applied peculiar modules theory for Floer homology of 4-ended tangles, using immersed curve interpretation.
result Corrected the rank of Knot Floer Homology for some pretzel knots.
Curve Shortening Flow preserves circularity for convex projections.
problem Understanding the behavior of curves under Curve Shortening Flow.
method Contradiction argument and analysis of tangent flows.
result Smooth curves with convex projections become asymptotically circular under Curve Shortening Flow.
Characterizes a specific type of convex curves on a 3-sphere.
problem Understanding convex curves on a 3-sphere.
method Decomposes curves on 3-sphere into 2-sphere curves, characterizes locally convex ones.
result Completely characterized a class of convex curves on the 3-sphere.
Holomorphic curves in moduli spaces are quasi-isometrically immersed.
problem Understanding the geometric properties of holomorphic curves in moduli spaces.
method Analyzing the quasi-isometric immersion of holomorphic maps from hyperbolic surfaces to moduli spaces.
result Holomorphic curves are quasi-isometrically immersed with parameters depending on surface and moduli space properties.
Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.
problem Investigating geometric properties of immersed curves with fractional Sobolev metrics.
method Analyzing Riemannian metrics on spaces of immersed curves, proving completeness and geodesic properties.
result Fractional Sobolev metrics are geodesically complete for .
A conformal immersion of a 2-torus into the 4-sphere is characterized by an auxiliary Riemann surface, its spectral curve. This complex curve encodes the monodromies of a certain Dirac type operator on a quaternionic line bundle associated to the immersion. The paper provides a detailed description of the geometry and …