Study on oriented disingquandles for distinguishing singular links.
problem Distinguishing singular links of different colors.
method Introduced oriented dichromatic singular links and oriented disingquandles to define invariants.
result Found invariants that can distinguish some singular links.
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. Paper extends group actions on Seifert manifolds with non-orientable base space.
problem Finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds with non-orientable base space.
method Constructing a double cover and isomorphism between fiber-preserving diffeomorphisms and actions on the cover.
result Construction of all actions satisfying a condition on the obstruction term and structure of finite groups.
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.
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.
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.
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…
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.
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.
The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.
problem Conditions for orientability in spaces with lower Ricci curvature bounds.
method Equivalent characterizations of orientability using Ricci limit and RCD spaces.
result Four-manifolds with Ricci curvature bounded below and volume non-collapsing are uniformly locally orientable.
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.
Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
problem Characterize fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.
method Construct a class of involutions, extend product involutions, and use double covering.
result Any fiber-preserving, orientation-reversing involution factors as a product of an orientation-preserving and a specific class of involutions.
Proves Alexander theorem for oriented Thompson group.
problem Establishing representation of oriented knots and links.
method Analogy with classical Alexander theorem for knots.
result Every oriented knot/link can be represented by an element of the oriented Thompson group.
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.
Orbifold local orientability can be detected by heat invariants.
problem Detecting local orientability in orbifolds.
method Using heat invariants to show Laplace isospectrality.
result Locally orientable orbifolds are distinguishable by heat invariants.
A method is described to consistently orient a specific 3D manifold.
problem Orienting the Heisenberg 3-manifold consistently.
method Described a canonical orientation method.
result A canonical orientation method for the Heisenberg 3-manifold is provided.
The paper generalizes conditions for Euler characteristic and orientability of manifolds.
problem Conditions for Euler characteristic and orientability of manifolds.
method Generalization of known conditions for Euler characteristic and orientability of manifolds.
result Even Euler characteristic for k-orientable manifolds unless dimension is a multiple of 2k+1. 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.
We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also associative. Along the way, we prove that there exists a tubular neighborhood N of every…
Defines orientation for metric measure spaces with Ricci curvature bounds.
problem Orientation of metric measure spaces with Ricci curvature bounds.
method Defined an orientation for measured Gromov-Hausdorff limit spaces of Riemannian manifolds with uniform Ricci bounds.
result Stability and uniqueness of orientations for noncollapsed sequences of Riemannian manifolds.
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
Researchers create a new compactification of character varieties using geometric and algebraic methods.
problem Compactifying character varieties of finitely generated groups in PSL2(R). method Geometric interpretation of elements of the real spectrum compactification as Γ-actions on R-trees, endowed with an orientation. result Continuous surjection from real spectrum compactification to oriented Gromov equivariant compactification.
DNNs generalize object recognition in novel orientations via neurons tuned to common features.
problem Understanding how DNNs generalize to objects in novel orientations.
method Training DNNs with familiar objects from multiple viewpoints and analyzing neuron responses.
result DNNs disseminate orientation-invariance from familiar objects to recognize objects in novel orientations.
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…
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.
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.
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…
Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.
problem Conditions for unknotting oriented virtual knots and links.
method Local moves (Delta, Sharp, Pass) for oriented virtual links.
result Lower bounds for the unknotting numbers are proven and shown to be best possible.
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 …
This paper adresses the following problem: Given a closed orientable three-manifold M, are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph manifolds. More presisely the main result of this paper asserts that any closed orien…
This study simplifies verification of invariants in oriented virtual knots.
problem Verifying invariants of oriented virtual knots is complex and time-consuming.
method Identifying a minimal generating set of oriented virtual Reidemeister moves.
result A four-element subset serves as a generating set for oriented virtual Reidemeister moves.
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.
The paper extends orientability results for DT invariants on Calabi-Yau 4-folds.
problem Defining Donaldson-Thomas invariants on quasi-projective Calabi-Yau 4-folds.
method Extending orientability results to quasi-projective Calabi-Yau 4-folds and using gauge theory.
result Orientability of moduli stacks and invariants on quasi-projective Calabi-Yau 4-folds.
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.
New combinatorial model for Milnor fibration using oriented matroids.
problem Understanding the homotopy type of Milnor fibers of complexified real arrangements.
method Introducing a poset quasi-fibration based on a subdivision of the Salvetti complex and an oriented matroid.
result Homotopy type of Milnor fiber depends only on the combinatorial structure of the oriented matroid.
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.
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.
Alternative proof and description of orientations for instanton moduli spaces.
problem Describing the relation between different orientations of G-instanton moduli spaces. method Using Witten localization of the index of a Dirac type operator.
result Alternative construction and proof of orientability of instanton moduli spaces.
A {\em word labeled oriented graph} (WLOG) is an oriented graph G on vertices X={x1,…,xk}, where each oriented edge is labeled by a word in X±1. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …
The study explores properties and mutations in oriented matroids, proving new results on Euclidean and non-Euclidean structures.
problem Investigating the Euclidean and non-Euclidean properties of oriented matroids.
method Analyzing the minimum number of mutations, using lexicographic extensions, and mutation-flips to prove properties.
result For rank 4 uniform oriented matroids, the minimum number of mutations adjacent to an element is at most 3.
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.
The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.
problem Defining and proving orientations for moduli spaces of geometric objects.
method Develops bordism categories and uses them to encode and solve orientation problems.
result Proves orientability and constructs canonical orientations for various moduli spaces.
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
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.
Method to create Heegaard splittings for graph manifolds.
problem Constructing Heegaard splittings of graph manifolds.
method Method to construct Heegaard splittings of oriented graph manifolds with orientable bases.
result A method to create Heegaard splittings of oriented graph manifolds.
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.