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.
New LP-algebras linked to surfaces, orientable and non-orientable.
problem Understanding LP-algebras through surfaces.
method Linking cluster algebras from surfaces to LP-algebras.
result Both orientable and non-orientable surfaces have associated LP-algebras.
J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…
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.
The paper defines H-orientability for surfaces in the Heisenberg group and finds non-H-orientable surfaces.
problem Defining orientability in the Heisenberg group for surfaces.
method Defined H-orientability for H-regular 1-codimensional surfaces in Hn. result Existence of non-H-orientable H-regular surfaces in H1. 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.
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.
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.
Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs …
Enhanced bikei modules distinguish unoriented and non-orientable surface-links.
problem Distinguishing unoriented and non-orientable surface-links.
method Extending biquandle module invariants to unoriented surface-links using bikei modules.
result Enhanced bikei modules are more effective at distinguishing non-orientable surface-links than bikei homset cardinality alone.
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.
Study stretch factors on nonorientable surfaces, proving nonorientable techniques ineffective.
problem Determine stretch factors on nonorientable surfaces.
method Analyzing pseudo-Anosov maps with orientable foliations on specific surfaces.
result Stretch factors do not have Galois conjugates on the unit circle.
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. 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. 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.
Invariants from biquasile colorings distinguish surface-links.
problem Counting and distinguishing surface-links.
method Coloring oriented surface-links using biquasiles and marked graph diagrams.
result Invariants can distinguish closed surface-links and cobordisms.
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 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.
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.
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.
We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via ano…
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.
Paper derives matrix formulae and proves skein relations for non-orientable surfaces in quasi-cluster algebras.
problem Understanding quasi-cluster algebras on non-orientable surfaces.
method Developed matrix formulae and proved skein relations for quasi-cluster variables.
result Laurent expansion and skein relations for quasi-cluster variables on 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 embedding surfaces into 3-manifolds, focusing on equivariant cases.
problem Embedding hyperbolic surfaces into hyperbolic 3-manifolds with specific symmetries.
method Examined orientation-preserving and orientation-reversing actions on surfaces, including nonorientable ones.
result Found conditions for equivariant embeddings of hyperbolic surfaces into hyperbolic 3-manifolds.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…
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.
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 …
Study surjections between braid groups on surfaces, focusing on lower central series.
problem Determine conditions for surjections between braid groups on surfaces.
method Utilizes properties of lower central series and combinatorial methods.
result Identifies specific values of m and n for which surjections exist between braid groups.
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus o…
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.
In this paper, the general formulation for inextensible flows of curves on oriented surface in R3 is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a partial differential equation involving the geodesic curvature and the geodesic…
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.
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.
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.
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.
Paper shows surfaces can't be heard to be orientable.
problem Determining orientability from spectral data.
method Applied Sunada's and Buser's methods to orbifolds.
result Constructed isospectral flat surfaces with different orientability.
Strong rigidity proven for non-compact surfaces.
problem Proving rigidity of non-compact surfaces.
method Generalized result showing proper homotopy to homeomorphism.
result All non-compact orientable surfaces, except plane and punctured plane, are topologically rigid.
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.
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. 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. Paper studies R∞-property for braid groups on surfaces, proving it for most cases.
problem Investigating the R∞-property for braid groups on orientable surfaces.
method Analyzing surface pure and full braid groups to determine the R∞-property.
result Most surface braid groups have the R∞-property, with exceptions noted.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
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.
Method calculates Z2-Thurston norm for Seifert 3-manifolds using pseudo-horizontal surfaces.
problem Computing Z2-Thurston norm for Z2-homology classes in orientable Seifert 3-manifolds. method Using pseudo-horizontal surfaces, we provide a necessary and sufficient criterion for their existence, calculate non-orientable genera, and develop an algorithm to compute the Z2-Thurston norm. result We successfully compute the Z2-Thurston norm for every Z2-homology class in orientable Seifert 3-manifolds. 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.