arXiv research

Study on oriented disingquandles for distinguishing singular links.

This paper extends the results from the author's previous paper to consider finite, fiber- and orientation- preserving group actions on closed, orientable Seifert manifolds $M$ that fiber over a non-orientable base space. An orientable base space double cover $\tilde{M}$ of $M$ is constructed and then an isomorphism be…

Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.

Analog of Kauffman bracket for non-orientable knots in thickened surface.

Study trisections of non-orientable 4-manifolds with boundary.

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 curves in non-orientable surfaces with specific intersection properties.

The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.

Study fiber-preserving, orientation-reversing involutions on Seifert fibered 3-manifolds.

Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.

This paper aims to define and study a notion of orientability in the Heisenberg sense ($\mathbb{H}$-orientability) for the Heisenberg group $\mathbb{H}^n$. In particular, we define such notion for $\mathbb{H}$-regular $1$-codimensional surfaces. Analysing the behaviour of a Möbius Strip in $\mathbb{H}^1$, we find a $1$…

Positive Thompson links are proven for oriented subgroup elements.

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…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

This paper calculates the non-orientable 4-genus for knots with 10 crossings.

DNNs generalize object recognition in novel orientations via neurons tuned to common features.

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…

The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.

Study virtualized Delta, Sharp, and Pass moves for oriented virtual knots and links.

We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length co…

In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, 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…

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 …

In this paper we deal with Seifert fibre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with boundary. Moreover, we compute a potentially sharp upper bound for their complexity…

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.

Polynomial-time algorithm finds short non-orientable loops intersecting graph edges up to 30 times.

New method finds non-orientable knotted surfaces in 4D.

New combinatorial model for Milnor fibration using oriented matroids.

Researchers refine the non-orientable $4$-genus of torus knots using Batson's surfaces.

The paper extends orientability results for DT invariants on Calabi-Yau 4-folds.

In this paper we define an orientation of a measured Gromov-Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncoll…

Minimal stretch factor for non-orientable surfaces is small.

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 $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immer…

Alternative proof and description of orientations for instanton moduli spaces.

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\cal G$ on vertices $X=\{ x_1,\ldots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm1}$. 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.

The book develops a new bordism-theoretic approach to understanding orientations of moduli spaces.

Enhanced bikei modules distinguish unoriented and non-orientable surface-links.

Minimal sets of moves for isotopic knots and trivalent graphs identified.

We show that the orientable double covering space of an indecomposable non-orientable $PD_3$-complex has torsion free fundamental group.

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…

Researchers compute bounds and formulas for non-orientable 4-genus of torus knots.

We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases.

Presentations for involutions on non-orientable surfaces up to genus 5.

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…

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifol…

Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.

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…