Adapts complex analytic methods to study non-orientable minimal surfaces.
problem Study of non-orientable conformal minimal surfaces in Rn. method Develops complex analytic methods from first principles.
result Proves real analytic Banach manifold for conformal minimal immersions.
We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An …
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. Minimal stretch factor for non-orientable surfaces is small.
problem Finding the minimal stretch factor for pseudo-Anosov homeomorphisms on non-orientable surfaces.
method Adapting Thurston's theory of fibered faces for non-orientable 3-manifolds.
result The minimal stretch factor is asymptotically on the order of 1/g.
Disproves polyhedral immersions of two specific triangulations on a non-orientable surface.
problem Proving non-existence of polyhedral immersions for triangulated surfaces.
method Developed method to disprove existence of polyhedral immersions in R^3.
result Two vertex-minimal, neighborly triangulations of a non-orientable surface are not realizable as polyhedral surfaces in R^3.
In this paper we give new existence results for complete non-orientable minimal surfaces in R3 with prescribed topology and asymptotic behavior.
Generators found for non-orientable surface mapping class group.
problem Finding minimal generating set for non-orientable surfaces.
method Szepietowski's system of generators, Dehn twists, and Y-homemorphisms.
result Szepietowski's system is a minimal generating set of mapping class group.
Paper finds a minimal set of Dehn twists to generate twist subgroup of non-orientable surfaces.
problem Finding a minimal set of Dehn twists to generate the twist subgroup of non-orientable surfaces.
method Used Dehn twists and an argument from Hirose [5] to find a minimal generating set.
result Found a generating set with one less element than the lower bound.
This paper calculates the non-orientable 4-genus for knots with 10 crossings.
problem Determining the non-orientable 4-genus for knots with a specific number of crossings.
method Calculating the minimal first Betti number of non-orientable surfaces smoothly embedded in a 4-ball with boundary the knot.
result The non-orientable 4-genus for knots with 10 crossings has been calculated.
A spine is constructed for a non-orientable surface's decorated Teichmüller space.
problem Constructing a spine for a non-orientable surface's decorated Teichmüller space.
method Building on Harer's work, constructing a spine and computing its dimension, showing equivariance with the pure mapping class group.
result A spine is constructed with minimal dimension for a punctured non-orientable surface.
We investigate constraints on embeddings of a non-orientable surface in a 4-manifold with the homology of M×I, where M is a rational homology 3-sphere. The constraints take the form of inequalities involving the genus and normal Euler class of the surface, and either the Ozsváth--Sazbó d-invariants or …
New recursion formula for non-orientable surfaces resolves divergences.
problem Computing volumes of moduli spaces for non-orientable surfaces.
method Generalization of Mirzakhani's recursion to non-orientable surfaces, handling divergences with integral kernels.
result Regularized volumes can be computed with a cutoff on crosscap size.
Study on minimal surfaces with closed curvature lines in 3D space.
problem Investigating complete non-orientable minimal surfaces with specific curvature properties.
method Analyzing complete non-orientable minimal surfaces of finite total curvature in R3 with ends foliated by closed lines of curvature. result There are no such surfaces with one end, proving a rigid situation.
The article improves and extends a heuristic for realizing surfaces with few vertices.
problem Realizing triangulated orientable and non-orientable surfaces with minimal vertices.
method Polyhedral embeddings and immersions of triangulated 2-manifolds with few vertices.
result Obtained numerous symmetric realizations of non-orientable surfaces with minimal vertices.
We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen …
We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g≥4.
Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles. For the smallest example of order 6 that is an ex…
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.
Proves minimal Heegaard surfaces properties in 3-manifolds.
problem Existence and properties of minimal Heegaard surfaces.
method Analyzes strongly irreducible Heegaard surfaces in 3-manifolds.
result Confirms conjecture about minimal surfaces' isotopy.
New stable minimal surfaces generalize classical Henneberg surface.
problem Finding new stable minimal surfaces in 3D.
method Generalized Henneberg surface with infinite families of complete, non-orientable surfaces.
result Infinite families of complete, finitely branched, non-orientable, stable minimal surfaces.
This paper proves hyperbolicity for curve graphs of non-orientable surfaces.
problem Proving hyperbolicity for curve graphs of non-orientable surfaces.
method Applying Hensel-Przytycki-Webb's argument for orientable surfaces to non-orientable surfaces.
result Curve graphs of non-orientable surfaces are 17-hyperbolic.
New method finds non-orientable knotted surfaces in 4D.
problem Reducibility of knotted surfaces in 4D.
method Elementary obstruction to reducibility.
result Construction of stably irreducible non-orientable surfaces.
We give a concrete example of an infinite sequence of (pn,qn)-lens spaces L(pn,qn) with natural triangulations T(pn,qn) with pn taterahedra such that L(pn,qn) contains a certain non-orientable closed surface which is fundamental with respect to T(pn,qn) and of minimal crosscap number among a…
The paper studies singularities in mean curvature flow and bounds on minimal surface total curvature.
problem Understanding singularities in mean curvature flow and bounds on minimal surface total curvature.
method Mean curvature flow and geometric analysis.
result The largest number k for which a minimal surface with total curvature less than k is a disk is greater than 3π.
Study on topological complexity and homotopy cofiber for non-orientable surfaces.
problem Understanding topological properties of non-orientable surfaces.
method Analysis of Lusternik-Schnirelmann category and topological complexity.
result Topological complexity of non-orientable surfaces of genus >3 is four.
We construct compact arbitrary Euler characteristic orientable and non-orientable minimal surfaces in the Berger spheres. Besides we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of …
Analog of Kauffman bracket for non-orientable knots in thickened surface.
problem Defining an invariant for non-orientable knots in a non-orientable surface.
method Proposes an analog of the Kauffman bracket polynomial with modified sign rules.
result Polynomial is an isotopy invariant and independent of classical Kauffman for orientable covers.
Presentations for involutions on non-orientable surfaces up to genus 5.
problem Representing involutions on non-orientable surfaces.
method Dehn twist--crosscap slide presentations.
result Presentations for involutions on non-orientable surfaces of genera up to 5.
We prove that closed surfaces of all topological types, except for the non-orientable odd-genus ones, can be minimally embedded in the Riemannian product of a sphere and a circle of arbitrary radius. We illustrate it by obtaining some periodic minimal surfaces in S2×R via conjugate constructio…
New invariant distinguishes non-orientable surfaces.
problem Distinguishing non-orientable surfaces bounded by the same knot.
method Mixed invariant from Lee and Bar-Natan deformations of Khovanov homology.
result Distinguishes exotic non-orientable surfaces.
Study on non-orientable surfaces for maximal disc packings.
problem Maximizing disc packings in non-orientable surfaces.
method Analyzing compact non-orientable surfaces of genus g≥3 for maximal k-packings. result Characterization of maximal disc packings in non-orientable surfaces.
Existence proved for metrics maximizing eigenvalue on non-orientable surfaces.
problem Finding metrics maximizing the first eigenvalue on non-orientable surfaces.
method Proved existence under spectral gap conditions.
result Existence of metrics maximizing eigenvalue on non-orientable surfaces.
Study extends surface embedding theorems to non-orientable cases.
problem Embedding non-orientable surfaces in 4-manifolds.
method Conditions for embedding non-orientable surfaces as sums of surfaces and projective planes.
result Extends Gabai and Auckly-Kim-Melvin-Ruberman-Schwartz theorems to non-orientable surfaces.
Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
Presented an infinite group presentation for non-orientable surfaces.
problem Presenting a group presentation for non-orientable surfaces.
method Used Dehn twists and crosscap pushing maps.
result An infinite presentation for the mapping class group of a non-orientable surface.
Presented an infinite group presentation for non-orientable surfaces.
problem Presenting the mapping class group of non-orientable surfaces.
method Generalized an existing presentation for mapping class groups.
result An infinite presentation for non-orientable surfaces with boundary.
Study on loops on non-orientable surfaces, determining cardinality and order.
problem Determining the cardinality and order of maximal complete 1-systems of loops on non-orientable surfaces.
method Proved the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is 2∣χ∣(∣χ∣+1), and used this to determine the cardinality of maximal complete 1-systems of loops. result Exact cardinality of maximal complete 1-systems of loops on punctured projective planes is determined.
Study homology groups for non-orientable surfaces with boundary.
problem Determine the first homology group for mapping class groups of non-orientable surfaces.
method Calculate the first homology group with twisted coefficients.
result Explicitly determined the first homology group for various mapping class groups.
The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surface bounded by the knot. We determine the crosscap numbers of torus knots.
This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.
problem Finite presentations for mapping class groups of non-orientable surfaces.
method Using Stukow's finite presentation and Birman exact sequences.
result An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
Study on dimensions of mapping class groups of non-orientable surfaces.
problem Determining dimensions of mapping class groups for non-orientable surfaces.
method Analyzing cohomological, geometric, and virtual dimensions.
result Equal dimensions for Ng when geq4,5. Study non-orientable surfaces, find finitely generated abelian subgroups.
problem Characterize abelian subgroups in non-orientable surfaces.
method Apply Birman-Lubotzky-McCarthy's arguments to non-orientable surfaces.
result Find a finitely generated group isomorphic to a given subgroup.
Paper finds a normal generating set for Torelli group of non-orientable surfaces.
problem Finding a normal generating set for the Torelli group of non-orientable surfaces.
method Explicit construction using BSCC and BP maps.
result An explicit normal generating set for the Torelli group of genus-g compact non-orientable surfaces with b boundary components. Finite rigid sets found in non-orientable surfaces.
problem Identifying rigid sets in non-orientable surfaces.
method Analyzing curve complexes for finite rigid sets.
result Finite rigid sets found for non-orientable surfaces.
Study Farrell cohomology for non-orientable surfaces, classifying subgroup conjugacy.
problem Determine the p-primary component of Farrell cohomology for non-orientable surfaces. method Classify subgroups of order p using topological equivalence adapted to surfaces with marked points. result Determine the p-primary component of Farrell cohomology for non-orientable surfaces. Triangle coordinates for integral laminations on non-orientable surfaces.
problem Describing integral laminations on non-orientable surfaces.
method Introducing triangle coordinates and providing an explicit bijection.
result Explicit bijection between integral laminations and a set of integer vectors.
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.
Classifies 3-manifolds from cube identifications.
problem Classifying non-orientable 3-manifolds.
method Identifying cube faces to form manifolds, classifying based on surface-complexity.
result Identifies four flat non-orientable 3-manifolds with surface-complexity one.