New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
A new method shows open Riemann surfaces have finite genus.
problem Determining the genus of open Riemann surfaces.
method Introducing a quotient space of forms to determine finite genus.
result Open Riemann surfaces have finite genus and can be embedded in a compact surface.
Study shows systole behavior changes significantly for large genus hyperbolic surfaces.
problem Understanding systole behavior in large genus hyperbolic surfaces.
method Analysis of random surfaces with respect to Weil-Petersson volume.
result Expected value of separating systole behaves like 2logg for large genus. Study on maximal surfaces with high genus in Lorentz-Minkowski space.
problem Existence of nonorientable maximal surfaces with high genus.
method Existence results for nonorientable maximal surfaces with high genus and one end.
result Existence of maximal surfaces with high genus in Lorentz-Minkowski space.
Algorithm calculates genus of embedded graphs on surfaces.
problem Finding minimal and maximal genus for graph embeddings.
method Constructing special branched coverings of the 2-sphere.
result Algorithm calculates orientable genus and maximal genus of graphs.
Proves prime theta-curves for knots on minimal genus surfaces.
problem Prime knots and essential arcs on Seifert surfaces.
method Analyzes prime knot unions with essential arcs on minimal genus surfaces.
result Each prime knot union an essential arc on a minimal genus Seifert surface is a prime theta-curve.
The paper finds infinite knots with surfaces of any positive genus.
problem Finding knots with surfaces of arbitrary positive genus.
method Constructing knot exteriors with longitudinal essential surfaces of any positive genus.
result Existence of infinitely many knots with surfaces of any positive genus.
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…
Study on invariant Seifert surfaces for strongly invertible knots, showing large gaps in genus.
problem Understanding gaps in genus between strongly invertible knots and their invariant Seifert surfaces.
method Analysis of invariant Seifert surfaces and proof of genus gaps, with variants of Edmonds' theorem.
result Gap between equivariant genus and usual genus can be arbitrarily large for strongly invertible knots.
Embedded surfaces in a ball have any genus and connected boundary.
problem Existence of embedded free boundary minimal surfaces with specific properties.
method Min-max techniques applied to the unit ball in R3. result Existence of embedded free boundary minimal surfaces with connected boundary and arbitrary genus.
Constructs a family of genus three minimal surfaces with parallel ends.
problem Creating embedded doubly periodic minimal surfaces with specific topological properties.
method Constructs a one-parameter family of surfaces with given Weierstrass data and solves the period problem.
result Solves the two dimensional period problem for the constructed surfaces.
New proof of high genus Lawson surfaces with area estimates.
problem Existence and area estimates of high genus Lawson surfaces.
method Deforming DPW potential to prove existence and calculating area.
result Estimates on area of Lawson surfaces in terms of genus.
Study cohomology of mapping class groups for big genus surfaces.
problem Characterize the first cohomology of pure mapping class groups for genus one and zero surfaces.
method Analyzes the first integral cohomology of pure mapping class groups for infinite type genus one surfaces and genus zero surfaces.
result For genus one surfaces, the first integral cohomology is trivial. For genus zero surfaces, there is an uncountable family of non-trivial homomorphisms to Z.
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
Classifies Teichmüller curves in genus three from genus two.
problem Classifying Teichmüller curves in genus three from genus two.
method Action on homology modulo two of affine groups of translation surfaces.
result Classification of orbits of square-tiled surfaces.
The study proves surfaces with high genus have a specific inequality.
problem Proving surfaces with high genus satisfy a specific inequality.
method Using volume entropy and systolic ratio inequality.
result Every closed surface of genus at least 18 satisfies Loewner's systolic ratio inequality.
We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of rational (or alternating) tangles in a projection of a link (or knot). For each g …
Researchers find the minimal genus for a specific type of surface bundle.
problem Determining the minimal genus of ΣgimesT2 bundles. method Constructing embedded surfaces to find exact minimal genus values.
result Exact values of minimal genus functions for ΣgimesT2 bundles are determined. For large genus hyperbolic surfaces, this paper proves eigenvalue conditions and diameter bounds.
problem Eigenvalue and diameter bounds for large genus hyperbolic surfaces.
method Analysis of moduli space of hyperbolic surfaces with Weil-Petersson metric.
result Generic hyperbolic surfaces of large genus have first eigenvalues greater than 3/16 - ε.
The paper calculates genus bounds for multibranched surfaces.
problem Finding genus bounds for multibranched surfaces.
method Using the first Betti number and boundary genus, the paper provides lower bounds for maximum and minimum genus.
result The maximum and minimum genus of GimesS1 equals twice that of G. The paper develops a theory for free boundary minimal surfaces with genus at least one.
problem Finding minimal surfaces with specific genus and boundary conditions.
method Using sweepouts of surfaces of genus g≥1 and m≥1 ideal boundary components, the paper constructs a min-max theory for free boundary minimal surfaces.
result The width for the area functional can be achieved by a bubble tree limit of branched genus g free boundary minimal surfaces with nodes.
The paper explores the genus of surfaces in complex projective spaces and improves minimal genus bounds.
problem Investigating the genus of surfaces in complex projective spaces.
method Analyzing knots and torus knots in CP2 and CP2#CP2. result The CP2-genus of knots is unbounded, unlike its topological counterpart. Study on spectral gaps of hyperbolic surfaces as genus increases.
problem Understanding differences in eigenvalues for large genus hyperbolic surfaces.
method Analysis of the Laplacian on degenerating hyperbolic surfaces, min-max principle.
result Supremum of spectral gaps has infimum limit of at least 1/4 as genus increases.
We give examples of isospectral non-isometric surfaces of genus 2 and 3 with variable curvatures and apply the result to construct isospectral potentials on Riemann surfaces of genus 2.
In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate wh…
Roberts proved that a family of alternating, arborescent, prime knots each have at least 22n−1 distinct minimal genus Seifert surfaces, where n is the genus of the knot in question. We give a subfamily of these knots that have exactly this many minimal genus Seifert surfaces.
This paper presents a method to obtain geometric registrations between high-genus (g≥1) surfaces. Surface registration between simple surfaces, such as simply-connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus t…
We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known surface for each genus. We illustrate multiple new examples for each genus g>2. In th…
Minimal surfaces in spheres found for any genus.
problem Finding minimal surfaces with arbitrary genus in 3-spheres.
method Topological structure analysis and embedding theorem.
result Every positive Ricci curvature 3-sphere contains a genus g surface.
Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.
problem Large geometric distortion when using spherical domain for genus-0 closed surfaces.
method Developed a novel method for ellipsoidal density-equalizing maps and combined with quasi-conformal maps.
result Significantly improved surface remeshing performance for genus-0 closed surfaces.
Algorithm decides if genus-two surfaces in 3-sphere are isotopic.
problem Determining isotopy of genus-two surfaces in S3. method 3-manifolds topology techniques, new algorithm for free groups.
result Algorithmic solution to genus-two surface isotopy problem.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
Formula proves Euler characteristic of singularized surfaces.
problem Calculating the Euler characteristic of singularized surfaces.
method Three operations: collapsing, zipping, and double loop identification.
result Formula for Euler characteristic of singularized surfaces.
Large PL surfaces in homology balls can have arbitrarily high genus.
problem Finding the minimum genus of PL surfaces in homology balls.
method Utilizes Heegaard Floer homology.
result The minimum genus can be arbitrarily large.
Flexible metrics found on a genus 2 surface.
problem Identifying non-rigid hyperbolic cone metrics on a genus 2 surface.
method Using a theorem by Erlandsson, Leininger, and Sadanand.
result Nine mapping class group orbits of non-rigid metrics found.
Researchers refine the non-orientable 4-genus of torus knots using Batson's surfaces.
problem Finding the minimum non-orientable 4-genus for torus knots. method Developed and analyzed Batson's non-orientable spanning surfaces in B4. result Batson's surfaces minimize the non-orientable 4-genus among certain surfaces. In 1841, Delaunay constructed the embedded surfaces of revolution with constant mean curvature (CMC); these unduloids have genus zero and are now known to be the only embedded CMC surfaces with two ends and finite genus. Here, we construct the complete family of embedded CMC surfaces with three ends and genus zero; the…
This paper studies the critical dynamics of random surfaces, focusing on area and genus evolution.
problem Understanding the time evolution of random surfaces and their genus.
method Analyzes the dynamics of area and genus using Cox-Ingersoll-Ross process and critical phenomena.
result The genus of surfaces evolves into two phases: planar surfaces and foamy surfaces.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
Two commutators generate mapping class groups of surfaces with genus ≥5.
problem Generating mapping class groups efficiently.
method Analyzing commutators in mapping class groups of surfaces.
result Mapping class groups of surfaces with genus ≥5 are generated by two commutators.
Complete minimal surfaces with any genus found in a specific 3-manifold.
problem Constructing minimal surfaces with arbitrary genus.
method Classical desingularization method.
result Complete embedded minimal surfaces of finite topology constructed.
New homotopy types defined for links in thickened surfaces with higher genus.
problem Defining stable homotopy types for links in surfaces with higher genus.
method Defined Khovanov-Lipshitz-Sarkar homotopy types and Steenrod squares for links in thickened surfaces with genus > 1.
result First meaningful Khovanov-Lipshitz-Sarkar stable homotopy types for links in 3-manifolds other than the 3-sphere.
The paper classifies group-actions on surfaces of small genus, focusing on bounding and geometrically bounding cases.
problem Classifying group-actions on surfaces of small genus, particularly focusing on bounding and geometrically bounding cases.
method Analyzing large group-actions on surfaces of genus 3, distinguishing between bounding and geometrically bounding cases.
result Identifies which large group-actions on surfaces of genus 3 are bounding or geometrically bounding.
Study how region crossing change behaves on surfaces of different genus.
problem Understanding region crossing change on surfaces of varying genus.
method Analyze (modified) region crossing change on higher genus surfaces.
result Behavior of region crossing change differs on higher genus surfaces.
Paper generalizes discrete uniformization for genus-zero surfaces.
problem Discrete uniformization for surfaces of genus zero.
method Reduction to planar cases via stereographic projections.
result Generalization of discrete uniformization to genus-zero surfaces.
New bounds on genus and area for CMC surfaces in 3-manifolds.
problem Bounding genus and area of CMC surfaces in 3-manifolds.
method Local degeneration of minimal surfaces and index-area bounds.
result Genus and area of CMC surfaces are bounded by index and area.
It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…