The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k k k -equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
Conditions for curves on a torus with specific pairwise intersections.
problem Finding curves on a torus with prescribed pairwise intersections.
method Necessary and sufficient conditions for curves on a torus with given pairwise intersections.
result Necessary and sufficient conditions for the existence of curves on a torus with specific pairwise intersections.
The study counts curves on a once-punctured torus with self-intersections.
problem Counting closed curves with self-intersections on a once-punctured torus.
method Combinatorial classification of curves with given word-length and self-intersections.
result Determination of curve counts with zero, one, and arbitrary self-intersections.
Study curves' intersections and distances, with applications in graph and group studies.
problem Understanding intersections and distances of curves.
method Using a relationship between intersection numbers and subsurface projection distances, applications in curve graphs and mapping class groups.
result Explicit quasi-constants for the relationship between intersection numbers and subsurface projection distances.
Proves curves on surfaces intersect at most once, matching known constructions.
problem Curves on surfaces intersecting at most once.
method Probabilistic argument in graph theory.
result Bound on cardinality of curves on surfaces.
The paper provides an algorithm to create curves touching a smooth cubic at specific intersection points.
problem Creating curves that touch a smooth cubic at specific intersection points.
method Algorithm based on divisions and Zariski tuples to produce n n n -contact curves. result An algorithm to generate n n n -contact curves to a smooth cubic. The study examines elastic curves with self-intersections and their properties.
problem Understanding self-intersecting elastic curves in Euclidean space.
method Review of classical elastica theory and discussion of open problems.
result Proof of Li--Yau type inequality for self-intersecting curves.
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
Study on intersection norms and disk complements of curves on surfaces.
problem Understanding intersection norms and their dual polytopes.
method Investigation of polytopes not dual to intersection norms and analysis of curve collections.
result Identification of polytopes not dual to intersection norms and study of curve collections.
Expository notes on intersection theory of holomorphic curves in 3-manifolds.
problem Intersection theory of punctured holomorphic curves in 3-manifolds.
method Minimal background in holomorphic curve theory, filled in appendices with proofs.
result Positive intersections and Siefring's intersection theory.
The paper finds the minimum number of intersections for curves under virtual homotopy.
problem Finding the minimum number of intersections for curves under virtual homotopy.
method Using generalizations of the Anderson--Mattes--Reshetikhin Poisson bracket and the Cahn cobracket.
result The minimal number of intersections equals to the number of terms of a generalized Poisson bracket for pairs of curves, and can be counted by a generalized cobracket for a single curve.
Algorithms compute geometric intersection numbers of curves efficiently.
problem Computing the minimal number of intersections of curves on surfaces.
method Simple algorithms for computing geometric intersection number, constructing curves, and deciding if intersections are zero.
result Efficient algorithms with polynomial time complexity for various curve intersection problems.
Study mapping class group orbits of curves with self-intersections on surfaces.
problem Counting and understanding orbits of curves with self-intersections on surfaces.
method Counting embeddings of ribbon graphs, asymptotic analysis, and counting curves with fixed minimal genus.
result Asymptotic number of orbits of curves with bounded self-intersections and minimal genus.
We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple curve, and that two simple curves on a surface with intersection number zero can be isotoped to be disjoint.
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…
Study curves in non-orientable surfaces with specific intersection properties.
problem Enumerate and understand curves in non-orientable surfaces with intersection constraints.
method Generalized construction of Malestein-Rivin-Theran to non-orientable surfaces.
result Lower bound for maximum number of curves in generic non-orientable surface.
Paper proves curves can be smoothed to reduce self-intersection by exactly 1.
problem Prove that the shortest closed geodesic self-intersects exactly k times.
method Carefully smoothing intersection points reduces self-intersection by exactly 1.
result The shortest closed geodesic self-intersects exactly k times for hyperbolic and Riemannian metrics.
Study detects if a circuit bounds a disc using curve intersections.
problem Determining if a circuit bounds an embedded disc.
method Analyzing the group generated by Dehn twists about curves in a circuit.
result Cycle relation between Dehn twists detects disc-boundability.
Constructs negatively curved complete intersections in complex manifolds.
problem Creating compact negatively curved complete intersections.
method Using Donaldson-Auroux theory to construct and prove existence.
result Existence of compact simply connected Kahler manifolds with negative holomorphic bisectional curvature.
The study constructs balanced and rigid curves on specific types of hypersurfaces and complete intersections.
problem Constructing balanced and rigid curves on Calabi-Yau and general-type complete intersections.
method Balanced and rigid curves are constructed using specific hypersurfaces and complete intersections.
result Rigid curves of various genera and balanced rational curves of high degrees are constructed.
The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…
Describes curves on surfaces with punctures and boundaries.
problem Representing multiple curves on surfaces with punctures and boundaries.
method Using geometric intersection numbers with embedded curves.
result Each multiple curve can be uniquely described.
Solves Plateau's problem for curves with self-intersections.
problem Plateau's problem for singular curves with self-intersections.
method Solution based on Plateau's problem for Jordan curves in metric spaces.
result New solution even in R n \mathbb{R}^n R n . We find bounds on non-simple closed geodesics on curved surfaces.
problem Finding bounds on non-simple closed geodesics on curved surfaces.
method Using bounds on length and self-intersection number.
result Exponentially tighter bounds on the number of non-simple closed geodesics.
Study of monodromy and vanishing cycles for complete intersection curves.
problem Computing topological monodromy of complete intersection curves.
method Innovative tools for studying monodromy of tensor products of very ample line bundles, induction on multi-degree.
result Answer given by the r-spin mapping class group associated to the maximal root of the adjoint line bundle.
Curve diffusion flow straightens curves with endpoints on intersecting lines.
problem Straightening open curves with endpoints on intersecting lines.
method Curve diffusion flow with mixed boundary conditions.
result The curve converges to a circular arc of the same length.
Paper calculates properties of curves in 5D space.
problem Calculating geometric properties of curves in 5D space.
method Algorithms for differential geometric properties and geodesic curvature.
result Geometric properties of transversal intersection curves in R^5.
Constructs infinitely many length equivalent curves using Goldman bracket.
problem Finding infinitely many length equivalent curves in a surface.
method Using the Goldman bracket between curves and their intersections.
result Constructs infinitely many pairs of length equivalent curves.
The study limits intersections of curves on a torus.
problem Bounding intersections of curves on a torus.
method Elementary, combinatorial, and geometric methods.
result The size of distinct homotopy classes of curves intersecting at most k times is k + O(\sqrt{k} \log k).
Mapping class group subgroups yield quasi-isometric curve complex.
problem Understanding the curve complex through coset intersections.
method Proving quasi-isometry and combinatorial equivalence of curve complex and coset intersection complex.
result Automorphism group of coset intersection complex is the extended mapping class group.
New method detects geometric intersection number greater than zero for curves on surfaces.
problem Detecting geometric intersection number greater than zero for curves on surfaces.
method Computing a value in the first homology group using elements of the fundamental group and Dehn twist.
result Explicit formula for Dehn twist action on free groups provides effective tool.
The study of smoothing arcs and curves on surfaces, proving tautness and arc length spectrum properties.
problem Analyzing the geometric and combinatorial effects of smoothing intersections in arcs or curves.
method Geometric and combinatorial analysis, proving tautness and arc length spectrum properties.
result Shortest arcs with self-intersections have exactly or at most one more self-intersection than the self-intersection number.
Study on curves on surfaces with bounds on intersections.
problem Lower bounds on curves filling punctured surfaces.
method Analyzes cardinality of curves intersecting at most k times, and uses systoles on hyperbolic surfaces.
result Establishes orders of growth for intersections, differing for even and odd k.
We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for…
Study provides bounds for longest curves with intersections on Teichmüller space.
problem Finding the longest curve with a given number of intersections on Teichmüller space.
method Uses lower bounds for the infimum of length functions associated with curve collections and dual cube complexes.
result Obtains estimates for the longest curve with k self-intersections.
Algorithm counts intersections of normal curves efficiently.
problem Efficiently solving word problems in mapping class groups of punctured surfaces.
method Fast algorithm for counting intersections of normal curves on triangulated surfaces.
result Efficient solution of the word problem for mapping class groups of punctured surfaces.
Study fills a surface with odd, non-3 curves.
problem Determining filling pairs for surfaces with odd, non-3 punctures.
method Constructed minimally intersecting pairs of curves.
result Completed the classification of filling pairs for all surfaces.
A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove a global analogue of this result, as follows. Suppose that two …
A classical inequality which is due to Lickorish and Hempel says that the distance between two curves in the curve complex can be measured by their intersection number. In this paper, we show a converse version; the intersection number of two curves can be measured by the sum of all subsurface projection distances betw…
Study of Hamiltonian flows on character varieties for self-intersecting curves.
problem Analyzing periodic orbits of Hamiltonian flows on character varieties.
method Explicit computations in Fock-Goncharov coordinates.
result Hamiltonian flows of trace functions associated to self-intersecting curves on a pair of pants have periodic orbits.
The intersection of certain submanifolds yields J-holomorphic curves.
problem Characterizing intersections of almost complex submanifolds.
method Using differential geometry of almost Hermitian manifolds to produce J-holomorphic curves.
result Degree one pseudoholomorphic maps between almost complex 4-manifolds are birational morphisms.
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new r r r -spin and Theta-class intersection numbers. Quadratic growth of intersecting curves on surfaces resolved.
problem Understanding the largest size of intersecting simple closed curves on surfaces.
method Introduced almost nibs, flowers, and stem systems to analyze curve intersections.
result The size of intersecting curves grows quadratically with the surface's Euler characteristic.
Study intersections of curves on translation surfaces, focusing on regular polygons and their Teichmüller disks.
problem Maximizing algebraic intersection between curves of given lengths.
method Investigate the quantity KVol defined for any closed orientable surface, focusing on regular n-gons for even n ≥ 8.
result Maximize algebraic intersection between curves of given lengths.
New bounds on curves on torus with few intersections.
problem Finding the maximum number of non-homotopic curves on a torus with limited intersections.
method Analyzing the maximum size of sets of curves with at most k intersections, using combinatorial optimization techniques.
result The maximum size of such a set is k+6 for all k, and k+4 for large k.
Study shows boundary curves of free boundary minimal surfaces are circles.
problem Characterizing boundary curves of free boundary minimal surfaces.
method Holomorphic techniques applied to the unit ball in Euclidean 3-space.
result Intersection curves of free boundary minimal surfaces with the unit sphere are all circles.
Sparse curves on surfaces grow at a specific intermediate rate.
problem Understanding growth patterns of sparse curve systems on surfaces.
method Analyzing the intersection numbers and sizes of sparse curve systems.
result Sparse curve systems grow roughly like c g c^{\sqrt{g}} c g . We present an approach of computing the intersection curve C \mathcal{C} C of two rational parametric surface § 1 ( u , s ) §_1(u,s) § 1 ( u , s ) and § 2 ( v , t ) §_2(v,t) § 2 ( v , t ) , one being projectable and hence can easily be implicitized. Plugging the parametric surface to the implicit surface yields a plane algebraic curve G ( v , t ) = 0 G(v,t)=0 G ( v , t ) = 0 . By analyzing the topology …