Simple lifts of non-simple curves on surfaces.
problem Finding simple lifts of non-simple closed curves on surfaces.
method Constructing finite covers and curves on surfaces.
result Explicit non-simple closed curves with simple lifts on degree n covers.
Characterizes covers using simple closed curves on surfaces.
problem Tackles the equivalence of covers via simple closed curves.
method Uses Teichmüller theory and the complex of curves.
result Two covers are equivalent if and only if the same curves lift to simple curves.
Mirzakhani extended curve counting from simple to all curves.
problem Counting curves in the mapping class group orbit with length at most L.
method Low-tech argument showing general result from simple curves.
result Derived the asymptotic growth of curves for arbitrary curves.
The paper characterizes simple closed curves on surfaces using profinite rigidity.
problem Characterizing simple closed curves on surfaces using profinite rigidity.
method Proving that elements with the same images under all finite groups are simple closed curves.
result The set of simple closed curves is closed in the profinite topology of the surface group.
Study on frequencies of non-simple curves in surfaces of large genus.
problem Frequency of non-simple curves in surfaces of large genus.
method Expression for frequency, large genus asymptotics, comparison with previous work.
result Identify most common types of non-simple curves with K intersections.
Undecidability found in surface curve existence problems.
problem Existence of simple closed curves in surface groups.
method Various questions about normal subgroups of surface groups.
result Undecidability of existence questions.
Algorithm decides if curves on a surface are simple.
problem Deciding if curves on a surface are simple.
method Combining TQFT representations and LERF for surface groups.
result Representation theoretic algorithm for deciding curve simplicity.
Curvature criteria for A-simple singularities and their parallel curves identified.
problem Determining singularity types of A-simple singularities and their parallel curves.
method Defined curvature parameters and criteria for A-simple singularities.
result Criteria to determine singularity types of A-simple singularities and their parallel curves.
Simple curve in 3D sums to cube, non-rectifiable.
problem Constructing a simple closed curve in 3D whose convex hull equals half-sum of the curve with itself.
method Constructing a specific curve in R^3 and proving its properties.
result A simple closed curve in R^3 exists whose convex hull equals half-sum of the curve with itself.
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
A loop group method classifies circles with simple factors.
problem Classifying circles with additional simple factors.
method Loop group description for curves in R^3.
result Classification of circletons: Circles dressed by simple factors.
Simple curve lifts don't generate all homology in finite covers.
problem Do lifts of simple curves generate the homology of finite covers?
method Examples provided to show not all simple curve lifts generate the homology.
result Not all simple curve lifts generate the homology of finite covers.
Simple curves enclose two small disks if they're wide and bend moderately.
problem Bounding the diameter of a curve to enclose two disjoint unit disks.
method Analyzing curvature and diameter constraints of a simple closed curve.
result A curve with curvature ≤1 and diameter ≥4 encloses two disjoint open unit disks.
Proves contractibility of a curve traced multiple times under certain length constraints.
problem Contractibility of a curve traced multiple times under length constraints.
method Analyzes simple closed curves on Riemannian manifolds and their homotopies.
result If a multiple-traced curve is contractible, the original curve is contractible under similar length constraints.
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-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.
Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simp…
Classifies curves on a punctured Klein bottle, providing a counterexample to a loop conjecture.
problem Classifying curves on a punctured Klein bottle and addressing the simple loop conjecture.
method Explicit description of the mapping class group and classification of fundamental group elements.
result Provides a counterexample to the simple loop conjecture for Klein bottle representations.
Simple closed curves in ε-boundaries separate sets in the plane.
problem Separating sets with simple closed curves in ε-boundaries.
method Analyzing ε-boundaries of planar sets and proving the existence of simple closed curves.
result Simple closed curves in ε-boundaries separate sets in the plane.
Homotopy types of curve and arc complexes are studied.
problem Understanding the homotopy types of curve and arc complexes.
method Proving homotopy equivalence and contractibility of complexes.
result Fine curve complex is homotopy equivalent to curve complex, fine arc complex is contractible.
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.
New findings on non-congruent curves with identical signatures.
problem Identifying congruence of non-degenerate curves with non-simple signatures.
method Associated directed graphs to signatures and used paths to reflect global and local symmetries.
result Non-congruent, non-degenerate curves can have identical signatures.
Conditions for simple closed curves in surface covers.
problem Conditions for simple closed curves in surface covers.
method Necessary and sufficient conditions for integral homology classes.
result Conditions for existence of simple closed curves in covers.
Characterizes Veech surfaces using lengths of simple closed curves.
problem Understanding Veech surfaces through curve lengths.
method Characterization via SL(2,R)-infimal lengths of simple closed curves.
result Veech surfaces characterized in terms of curve lengths.
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
New curves can converge to any geodesic on a sphere.
problem Existence of unique geodesic limits for curve shortening flows.
method Constructed a smooth metric on S^2 and a flow converging to multiple geodesics.
result Multiple geodesics can be limits of a curve shortening flow.
Automorphisms of fine 1-curve graph linked to surface homeomorphisms.
problem Understanding automorphisms of fine 1-curve graphs.
method Isomorphic mapping to surface homeomorphisms.
result Automorphism group is isomorphic to homeomorphism group of a surface.
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.
Study on moduli spaces of sextic curves with simple singularities and their compactifications.
problem Understanding moduli spaces of sextic curves with simple singularities.
method Using period maps of K3 surfaces with ADE singularities, algebraic open embeddings into arithmetic quotients of type IV domains, and GIT and Looijenga compactifications.
result Identifications of GIT and Looijenga compactifications for all cases.
Study earthquake deformations on a once-punctured torus.
problem Understanding earthquake deformations on Teichmüller space.
method Two methods: linear recurrence relations and hyperbolic geometry.
result Algebraic and geometric interpretations of earthquake deformations.
The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.
In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0…
We construct simple curves from immersed curves in the setting of handlebodies and Heegaard splittings. We define a measure of complexity we call girth for closed curves in a handlebody. We extend this complexity to Heegaard splittings and pose a conjecture about all Heegaard splittings. We prove a test case of this co…
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.
We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to certain simple moves in which only one curve changes, and whose 2-cells correspo…
In this note, we develop a condition on a closed curve on a surface or in a 3-manifold that implies that the curve has the property that its length function on the space of all hyperbolic structures on the surface or 3-manifold completely determines the curve. For an orientable surface S of negative Euler characteris…
Characterizes unknotted curves on Seifert surfaces of twist knots.
problem Identifying unknotted curves on Seifert surfaces of twist knots.
method Analyzing homologically essential simple closed curves on Seifert surfaces of genus one knots.
result Characterizes unknotted curves on Seifert surfaces of twist knots, including infinitely many for the figure eight knot and one for Whitehead doubles.
Random simple closed curves map Teichmüller space to geodesic currents.
problem Mapping Teichmüller space to geodesic currents.
method Using a formula for intersection numbers of multicurves and Dehn coordinates.
result Proper embedding of Teichmüller space into the space of geodesic currents.
Given a pair of curves C_1 and C_2 on a hyperbolic surface F, when does there exist a pseudo-Anosov map sending one to another? More generally, one may ask the same question for C_i to be sets of disjoint simple closed curves. We will give necessary and sufficient conditions for the existence of such maps.
We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.
Goldman and Turaev found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinator…
Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
Survey on geodesics on tetrahedra in curved spaces.
problem Understanding geodesics on tetrahedra in curved spaces.
method Analyzing geodesics on regular tetrahedra in spaces of constant curvature.
result Results on the behavior of simple closed geodesics.
Let Ngk be a nonorientable surface of genus \ g≥5 \ with \ k-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on Ngk. Along the way, we will fill some little gaps in the proofs of some theorems in \cite{A} and \cite{I1} giving algebraic char…
Characterizes when curves form bouquets in surfaces.
problem Understanding when simple closed curves form bouquets in surfaces.
method Characterization in terms of relations between Dehn twists.
result Characterization of bouquets of curves in surfaces.
We provide new results and new proofs of results about the torsion of curves in R3. Let γ be a smooth curve in R3 that is the graph over a simple closed curve in R2 with positive curvature. We give a new proof that if γ has nonnegative (or nonpositive) torsion, then γ has zero …
Globally irreducible nodes (i.e. nodes whose branches belong to the same irreducible component) have mild effects on the most common topological invariants of an algebraic curve. In other words, adding a globally irreducible node (simple nodal degeneration) to a curve should not change them a lot. In this paper we stud…
Alternative approach to curve shortening flow solutions.
problem Understanding singularity formation in mean curvature flow.
method Simple ODE description and alternative proof of self-similar solutions.
result Self-similar solutions lie in planes.