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.
In a paper published in 2002, the author gave a criterion to determine whether there is a fiber-preserving branched covering between two given orientable Seifert manifolds with orientable bases. Here we supply some details of the proof of two claims in that paper. We give an explicit construction of fiber-preserving br…
The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.
problem Identifying exceptions to fiber-preserving symmetry in ODEs and systems.
method Lie's classification of Lie algebras of vector fields, absolute and relative scalar differential invariants, conditional and vector-valued relative invariants, prolongations of actions.
result Examples of scalar ODEs and systems with symmetry groups not fiber-preserving.
Paper solves equivalence problems for fifth-order differential operators using Cartan's method.
problem Equivalence problem for fifth-order differential operators under fiber-preserving transformations.
method Cartan's method of equivalence applied to solve two versions of the equivalence problem.
result Sufficient and necessary conditions for fiber-preserving transformations between fifth-order differential operators.
The study explores finite group actions on elliptic 3-manifolds.
problem Finite, fiber-preserving group actions on elliptic 3-manifolds.
method General construction and restriction to elliptic 3-manifolds, proof for orientation-reversing and fiber-preserving diffeomorphisms, analysis of each type of elliptic 3-manifold.
result A presentation for the quotient space under finite group actions is constructed.
This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence pro…
In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem i…
Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifol…
Study focuses on classifying special geometric structures.
problem Classify singular affine structures of integrable systems.
method Classification through simple semitoric systems equivalence.
result Counterexamples exist for multiple pinched fibers.
Maps between circle bundles are studied, proving fiber-preserving and finiteness results for mapping degrees.
problem Understanding the properties of maps between circle bundles and their degrees.
method Analyzing the structure of circle bundles over aspherical manifolds and using homotopy and homology properties.
result The mapping degree set of fiber-preserving maps from E1 to E2 is determined and finite under certain conditions. We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.
This paper is devoted to apply the equivariant moving frame method to study the local equivalence problem of third order ordinarily differential equation under the pseudo-group of fiber preserving transformations.
Study local equivalence of Riemannian submersions using differential invariants.
problem Local equivalence problem for Riemannian submersions under fiber-preserving isometries.
method Analysis of differential invariants for orbit submersions induced by a Killing field.
result Explicit formulas for A and H in terms of base data (gˉ,φ,Ω) and equivalence criterion. Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
Study classifies Calabi-Yau threefolds with non-Gorenstein involutions.
problem Understanding non-Gorenstein involutions on Calabi-Yau threefolds.
method Classification of Calabi-Yau threefolds with specific properties.
result Classification of Calabi-Yau threefolds with Picard rank one and non-Gorenstein involutions.
The paper studies circular evolutes and involutes of framed curves in Euclidean space.
problem Investigating properties of framed curves and their evolutes and involutes.
method Definition and analysis of circular evolutes and involutes of framed curves, properties of normal surfaces, and their relations.
result Circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similar to fronts in the Euclidean plane.
New formula for dual knots using involutions.
problem Understanding dual knots and their transformations.
method Involutive analog of knot surgery formula.
result Computed local equivalence class for involutive dual knots.
The study classifies involutions on del Pezzo surfaces.
problem Classifying involutions on del Pezzo surfaces.
method Mapping class group theory and hyperbolic reflection groups.
result A complete classification of involutions on del Pezzo surfaces.
Characterizes the Legendre involution on generic frontals.
problem Identifying the Legendre involution on a specific class of frontals.
method Analyzes generic frontals under mild assumptions and uses complexification.
result Any involution with the same fixed points as the Legendre involution is the Legendre involution.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
Study proves naturality and functoriality in a type of Heegaard Floer homology.
problem Proving naturality and functoriality in a specific type of Heegaard Floer homology.
method Used the doubling model for the involution and variations to prove results.
result First-order naturality of involutive Heegaard Floer homology proved.
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f, which is a lower bound for the number of fixed points among all maps homotopic to f. The main theorem of this paper proves this conjecture for all orientation pre…
The paper develops a new theory for knots and 3-manifolds with involutions.
problem Developing a new theory for knots and 3-manifolds with involutions.
method Establishing a version of Seiberg-Witten Floer K-theory for knots and 3-manifolds with involutions.
result 10/8-type inequalities for knots and involutions, yielding lower bounds on stabilizing numbers and relative genera.
The study proves symplectic quandles cannot have good involutions.
problem Existence of good involutions in symplectic quandles.
method Investigation of necessary and sufficient conditions for good involutions.
result Nonexistence of good involutions in symplectic quandles.
Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
Minimal involutions generate a subgroup of nonorientable surfaces.
problem Generating a minimal set of involutions for a specific subgroup.
method Obtained a minimal generating set of involutions.
result Minimal involutions for the level 2 subgroup of a nonorientable surface.
Study exact surgery formula in involutive Heegaard Floer homology.
problem Understanding integer homology spheres through knot surgery.
method Using doubling model of involution and mapping cone formula.
result Examples of non-homology cobordant integer homology spheres.
We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all n⩾3 PU(n,1) has involution length at most 8.
Anti-symplectic involutions connect a sphere in a symplectic surface.
problem Understanding involutions on Lagrangian spheres in symplectic quadrics.
method Using Hamiltonian isotopy to show connections between involutions.
result Anti-symplectic involutions are Hamiltonian isotopic.
Real slices of parabolic opers on Riemann surfaces are studied.
problem Understanding the fixed-point locus of involutions on parabolic opers.
method Investigated the space of parabolic SL(r,C)-opers and their involutions.
result Fixed-point loci of involutions on different descriptions of parabolic opers coincide.
Classifies periodic diffeomorphisms on surfaces commuting with specific involutions.
problem Classifying periodic automorphisms on surfaces that commute with certain involutions.
method Analyzes irreducible periodic automorphisms on surfaces Σg that commute with hyperelliptic involutions. result A classification up to conjugacy for irreducible periodic automorphisms of a surface Σg commuting with involutions ι such that Σg/⟨ιangle is homeomorphic to T2. Computed involutive knot invariants for specific pretzel knots.
problem Computing involutive knot invariants for a specific class of knots.
method Computed involutive knot invariants for pretzel knots of the form P(-2,m,n) with m and n odd and ≥ 3.
result Computed involutive invariants for a specific class of knots.
Paper studies involutions generating the twist subgroup of nonorientable surfaces.
problem Generating the twist subgroup by involutions on nonorientable surfaces.
method Analyzes involutions to find the smallest generating sets.
result Provides generating sets of involutions with minimal elements.
Let Σg,b denote a closed orientable surface of genus g with b punctures and let Mod(Σg,b) denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, Mod(Σg,b) is generated by involutions. He also asked if there exists a universal upper bound, indepe…
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.
Three involutions generate the mapping class group for surfaces of genus 6 or more.
problem Generating the mapping class group with minimal involutions.
method Proving the group is generated by three involutions for surfaces of genus 6 or more.
result The mapping class group is generated by three involutions for surfaces of genus 6 or more.
In this article, we classify all involutions on S^6 with 3-dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted 3-sphere and the classification of free involutions on homotopy CP^3's.
The paper defines evolutes and involutes for framed curves and their properties.
problem Defining evolutes and involutes for framed curves with singular points.
method Using the theory of framed curves and Bertrand type curves.
result Conditions for evolutes and involutes being inverse operations of framed curves.
This thesis proves how to generate a specific group using involutions.
problem Generating the mapping class group of non-orientable surfaces by involutions.
method Using a set of involutions, the thesis provides the number of elements needed for generation based on the surface's genus and punctures.
result The mapping class group of non-orientable surfaces can be generated by a fixed number of involutions, independent of the surface's genus and punctures.
We remark some basic facts on homological aspects of involutive Lie bialgebras and their involutive bimodules, and present some problems on surface topology related to these facts.
Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and …
The study describes good involutions in quandles and Alexander quandles.
problem Characterizing and enumerating good involutions in quandles and Alexander quandles.
method Completely describing good involutions of free and subquandles of twisted conjugation quandles of groups, including Alexander quandles.
result Explicit mappings for good involutions of linear quandles up to order 23.
Defines horocyclic evolutes, parallels, and involutes of spacelike frontals in hyperbolic 2-space.
problem None explicitly stated; focuses on definitions and relations.
method Using enveloid theorem, defines horocyclic parallel and involute as normal envelopes of horocycles.
result Investigates relations among horocyclic evolutes, parallels, and involutes.
New model improves MCMC efficiency and multi-modal distribution exploration.
problem Inefficient and slow MCMC methods for complex distributions.
method Deep involutive generative models for Metropolis-Hastings updates.
result Deep involutive models can learn complex MCMC updates efficiently.
Proposes a new notation for biracks to simplify rack structure analysis.
problem Simplifying birack notation for easier rack structure analysis.
method Introduces a new notation that includes rack structure knowledge from the start.
result Generalizes results from involutive to non-involutive biracks and clarifies rack structure relations.
We study the orientation preserving involutions of the orientable 3-dimensional handlebody Hg, for any genus g. A complete classification of such involutions is given in terms of their fixed points.
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an …