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.
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.
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.
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.
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.
The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in Rn for any n≥3. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…
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.
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.
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.
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 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 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 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.
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 …
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.
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. 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.
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.
We show that the Lusternik-Schnirelmann category of the homotopy cofiber of the diagonal map for non-orientable surfaces equals three. Also, we prove that the topological complexity of non-orientable surfaces of genus >3 is four.
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.
The non-orientable 4-genus of a knot in the 3-sphere is defined as the smallest first Betti number of any non-orientable surface smoothly and properly embedded in the 4-ball, with boundary the given knot. We compute the non-orientable 4-genus for all knots with crossing number 8 or 9. As applications we prove a conject…
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 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.
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 Euler class and flux homomorphisms for non-orientable surfaces.
problem Investigate Euler class and flux homomorphisms for non-orientable surfaces.
method Analyze Euler class and flux homomorphisms for non-orientable compact surfaces with one boundary component.
result Prove the simplicity of the kernel of the flux homomorphisms, implying the non-existence of invariants analogous to the Calabi invariant.
The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.
problem Existence of positive scalar curvature metrics on non-orientable manifolds and their covers.
method Extends Schoen-Yau inductive descent approach to non-orientable manifolds.
result Examples of non-orientable manifolds with positive scalar curvature metrics on their orientation double covers but not on homotopy equivalent manifolds.
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.
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 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.
Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.
problem Defining an invariant for pseudo-classical knots in non-orientable thickening of a non-orientable surface.
method Introducing an invariant Δ that is an analogue of Turaev comultiplication, defined in terms of homotopy classes of loops on the surface. result Analogous invariants to affine index polynomial for pseudo-classical knots in non-orientable manifolds.
Study curvatures of diffeomorphisms on non-orientable surfaces.
problem Computing curvatures of measure-preserving diffeomorphisms on non-orientable surfaces.
method Extending Arnold and Lukatskii's approach, computing curvatures and asymptotics.
result Computed curvatures and asymptotics for the Klein bottle and real projective plane.
Study on non-orientable hyperbolic 3-manifolds and their deformations.
problem Understanding the deformation space of non-orientable hyperbolic 3-manifolds.
method Computing the deformation space of pairs (M^3, Δ) and determining representations in Isom(H^3).
result Existence of deformations not realizable as pair deformations.
In the symplectization of standard contact 3-space, R×R3, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus 0. We show that any Legendrian knot has a non-or…
We show that the orientable double covering space of an indecomposable non-orientable PD3-complex has torsion free fundamental group.
Characterizes closures of mapping class group orbits on non-orientable surfaces.
problem Understanding closures of orbits in Teichmüller spaces for non-orientable surfaces.
method Analyzes closures in ML and PML for measured laminations, projective measured laminations, and points. result Characterizes closures of weighted two-sided curves in ML. Study shows non-orientable manifolds restrict signature-changing metrics globally.
problem Global obstructions to signature-changing metrics on non-orientable manifolds.
method Explicit geometric constructions based on Möbius strip topology.
result Radical of signature-changing metrics cannot be everywhere transverse.
We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the inf…
Topology of non-orientable spaces without boundary is studied.
problem Topology of non-collapsed RCD spaces without boundary.
method Studied the stability of non-orientability and topology under Gromov-Hausdorff convergence.
result Non-orientable spaces without boundary have a stable ramified double cover.
Counting lattice points in moduli space of Klein surfaces.
problem Count lattice points in moduli space of Klein surfaces.
method Introduced metric Möbius graphs, counted lattice points weighted by non-orientability measure, deduced recursion for volumes.
result Proved refined version of Norbury's recursion and computed refined Euler characteristic.
Study non-orientable link cobordisms using Floer homologies to prove inequalities.
problem Prove inequalities involving Euler characteristic and local maxima in non-orientable cobordisms.
method Use unoriented instanton and knot Floer homology to introduce unoriented versions of band unknotting number and refined cobordism distance.
result Show that the difference between unoriented refined cobordism distance of a knot from the unknot and non-orientable slice genus can be arbitrarily large.