Study introduces non-orientable 3-submanifolds in G2-manifolds.
problem Characterizing non-orientable three-submanifolds in G2-manifolds.
method Analogy with associative and co-associative cases, Cartan-Kähler theory.
result Some three-manifolds can be modeled on planes in a special G2-orbit.
Study uses Newton polytopes to distinguish Lagrangian fillings of Legendrian submanifolds.
problem Distinguishing Lagrangian fillings of Legendrian submanifolds.
method Utilizes Newton polytopes associated with augmented values of Reeb chords.
result Newton polytopes can distinguish infinitely many distinct Lagrangian fillings.
We study the Birman exact sequence for compact 3--manifolds, obtaining a complete picture of the relationship between the mapping class group of the manifold and the mapping class group of the submanifold obtained by deleting an interior point. This covers both orientable manifolds and non-orientable ones.
A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the o…
Study on Hamiltonian stationary cones with isotropic links in 5-dimensional complex space.
problem Characterizing properties of Hamiltonian stationary isotropic surfaces and submanifolds.
method Analyzing the geometry and topology of cones formed by isotropic submanifolds in complex spaces.
result Closed oriented immersed isotropic surfaces in S5 are either Legendrian and minimal or have specific Legendrian points. Study trisections of non-orientable 4-manifolds with boundary.
problem Understanding trisections in non-orientable 4-manifolds.
method Introduced trisections of non-orientable 4-manifolds with boundary, proved a non-orientable analogue of a theorem, and discussed adaptation of trisection theory.
result Existence of trisection diagrams and Kirby diagrams for closed non-orientable 4-manifolds.
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.
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.
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.
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.
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.
We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We appl…
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.
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. 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.
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.
Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.
problem Measuring the minimum genus of non-orientable surfaces bounded by torus knots.
method Computed bounds and provided a generalized formula for non-orientable 4-genus of torus knots.
result Computed bounds and a generalized formula for non-orientable 4-genus of torus knots.
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.
Polynomial-time algorithm finds short non-orientable loops intersecting graph edges up to 30 times.
problem Finding short non-orientable loops intersecting graph edges efficiently.
method Combining computational biology techniques with recent graph theory results.
result Existence of short canonical non-orientable systems of loops.
Researchers calculate the non-orientable 4-genus for knots with 8 or 9 crossings.
problem Determining the minimum non-orientable surface complexity for knots with specific crossing numbers.
method Computed the non-orientable 4-genus for knots with 8 or 9 crossings using surface embeddings in 4-ball.
result Computed the non-orientable 4-genus for all knots with 8 or 9 crossings, proving conjectures and providing new slicing number bounds.
New method for linking numbers in non-orientable 3D shapes.
problem Linking numbers in non-orientable 3-manifolds require orientation, which is not always available.
method Developed homotopy invariants to calculate linking numbers without orientation.
result Successfully calculated linking numbers in non-orientable 3-manifolds.
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.
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.
Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.
problem Understanding achiral Lefschetz fibrations from non-orientable ones.
method Composition of standard orientation double covering map and non-orientable Lefschetz fibration.
result Specification of a monodromy factorization for the composition.
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.
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.
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 non-orientable 4-genus for 11-crossing non-alternating knots.
problem Computing the non-orientable 4-genus for specific knots.
method Survey tools and use various techniques to calculate the invariant.
result Calculate non-orientable 4-genus for 11-crossing non-alternating knots.
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.
Study shows non-equivariant and equivariant non-orientable 4-genus of periodic knots can differ.
problem Exploring differences in non-orientable 4-genus for periodic knots.
method Analyzed p-periodic knots, showing differences in equivariant and non-equivariant non-orientable 4-genus.
result Differences exist in non-equivariant and equivariant non-orientable 4-genus for periodic knots.
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.
Non-orientable Lagrangian cobordisms exist for Legendrian knots, differing from orientable ones.
problem Understanding non-orientable Lagrangian cobordisms for Legendrian knots.
method Analyzing symplectization of standard contact 3-space and using exact, non-orientable Lagrangian cobordisms.
result Existence of non-orientable Lagrangian endocobordisms with crosscap genus being a positive multiple of 4.
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.
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.
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 on Legendrian knots and their non-orientable Lagrangian fillings.
problem Conditions for Legendrian knots to have non-orientable exact Lagrangian fillings.
method Developed combinatorial obstructions and classified fillability for various knot families.
result Completely determined decomposably non-orientable fillability for alternating and plus-adequate knots.
Corrects classification of Seifert fibrations for lens spaces with non-orientable bases.
problem Classifying Seifert fibrations of lens spaces with non-orientable bases.
method Corrected an earlier lemma and filled a classification gap.
result Corrected the classification of Seifert fibrations for lens spaces with non-orientable bases.
Paper describes Seifert fibre spaces, focusing on non-orientable cases.
problem Understanding the complexity of non-orientable Seifert fibre spaces.
method Combinatorial description and complexity bounds computation.
result Sharp upper bounds for complexity in non-orientable cases.
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 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 shows infinite-dimensional third bounded cohomology for non-orientable surfaces.
problem Understanding the third bounded cohomology of non-orientable surfaces.
method Analyzing measure-preserving homeomorphisms of non-orientable surfaces.
result Third bounded cohomology is infinite-dimensional.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
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 Whitehead doubles and their sliceness properties.
problem Understanding sliceness of Whitehead doubles of knots.
method Survey of techniques to obstruct sliceness and improve bounds on non-orientable genus.
result Improved bounds on non-orientable 4 genus of Whitehead doubles and genus 1 non-orientable cobordisms to cable knots.