Every negative amphichiral knot is rationally slice.
problem Proving every negative amphichiral knot is rationally slice.
method Systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior.
result Every negative amphichiral link is rationally slice.
The paper explores symmetric representations of links and conditions for amphichirality.
problem Investigating symmetric representations of links and conditions for amphichirality.
method Using antipodally self-dual and antipodally symmetric maps, the authors provide sufficient combinatorial conditions for amphichirality.
result A link is amphichiral if its self-dual pairing is not one of 6 specific ones.
Study proves obstructions to equivariantly slice strongly negative amphichiral knots.
problem Proving obstructions for equivariantly slice strongly negative amphichiral knots.
method Using determinant, Spinc-structures, Donaldson's theorem, and Heegaard Floer correction terms.
result 8 out of 16 strongly negative amphichiral knots with 12 or fewer crossings are not equivariantly slice.
We give a simple obstruction for a knot to be amphichiral, in terms of the homology of the 2-fold branched cover. We work with unoriented knots, and so obstruct both positive and negative amphichirality.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Study on equivariant Q-sliceness for strongly invertible knots.
problem Understanding Q-sliceness for strongly invertible knots.
method Constructive and obstructive approaches using Fox-Milnor condition and equivariant concordance.
result Klein amphichiral knots are equivariant Q-slice in a single Q-homology 4-ball.
Tanaka shows amphichiral symmetric unions of the unknot are trivial.
problem Understanding amphichiral symmetric unions and their Jones polynomials.
method Analyzing the Jones polynomial of amphichiral symmetric unions of the unknot and generalizing to other knots.
result Amphichiral symmetric unions of any knot with one twist region are trivial.
Research classifies knots based on sliceness and amphichirality.
problem Classifying odd-stranded Turk's head knots based on sliceness and amphichirality.
method Constructing commuting pairs of ambient involutions and analyzing the equivariant Fox-Milnor square condition.
result Established a sharp parity dichotomy for equivariant rational sliceness and Klein amphichirality of odd-stranded Turk's head knots.
New knots found that are 4-genus minimal.
problem Finding knots with minimal 4-genus.
method Constructing infinitely many amphichiral knots with specific properties.
result Knots with 4-genus minimal for each g>0. New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball VK. result Single 4-manifold accounts for all known rationally slice knots.
Paper defines half-Conway polynomial and computes it for knots up to 12 crossings.
problem Computing and characterizing half-Conway polynomials of knots.
method Normalized Conway polynomial, equivariant skein relation, diagrammatic interpretation.
result First examples of non-slice strongly negative amphichiral knots with determinant one.
Survey on embedding 3-manifolds in definite 4-manifolds, focusing on Donaldson's theorem.
problem Understanding embeddings of 3-manifolds in definite 4-manifolds.
method Utilizes Donaldson's diagonalization theorem and combinatorics of integral lattices.
result New result on embedding amphichiral lens spaces in negative-definite manifolds.
Using computational techniques we tabulate prime knots up to five crossings in the solid torus and the infinite family of lens spaces L(p,q). For these knots we calculate the second and third skein module and establish which prime knots in the solid torus are amphichiral. Most knots are distinguished by the skein mod…
In 1997 Cochran-Orr-Teichner introduced a natural filtration, called the n-solvable filtration, of the smooth knot concordance group, C. Its terms {F_n} are indexed by half integers. We show that each associated graded abelian group G_n=F_n/F_{n.5}, n>1, contains infinite linearly independent sets of elements of order …
We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-c…
Classifies symmetries of knots using group actions and orthogonal representation theory.
problem Classifying symmetries of knots in 3-sphere.
method Using geometrization and orthogonal representation theory, constructing examples, and distinguishing symmetries.
result Cyclic and dihedral families of symmetries of prime knots and composite knots.
We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion wh…
It is a natural consequence of fundamental properties of the Casson invariant that the Rokhlin invariant of an amphichiral integral homology 3-sphere M vanishes. In this paper, we give a new direct proof of this vanishing property. For such an M, we construct a manifold pair (Y,Q) of dimensions 6 and 3 equipped with so…
Survey of Turk's head knots and links properties.
problem Characterize Turk's head knots and links.
method Discussion of various results in the mathematical literature.
result Turk's head links are alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime.
The paper classifies involutions on S^4, proving linearities under certain conditions.
problem Classifying involutions on S^4 with specific fixed-point sets.
method Combining surgery theory, Schoenflies theorem, and equivariant topology.
result Linear involutions on S^4 with 1-dimensional fixed-point sets are proven.
The article explores the mapping class group using unicellular maps and provides filtrations.
problem Understanding the structure of the mapping class group.
method Using unicellular maps and surgeries, the article describes the mapping class group.
result Provides filtrations of the mapping class group.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
Maps are an important medium that enable people to comprehensively understand the configuration of cultural activities and natural elements over different times and places. Although massive maps are available in the digital era, how to effectively and accurately access the required map remains a challenge today. Previo…
The paper constructs biharmonic maps between spheres using polynomial maps.
problem Creating biharmonic maps between spheres.
method Using harmonic homogeneous polynomial maps of different degrees to generate proper biharmonic maps.
result Established a method for constructing proper biharmonic product maps.
Both bi-harmonic map and f-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f-bi-harmonic maps as the critical points of the f-bi-energy functional 21∫Mf∣τ(φ)∣2dvg. This class of maps generalizes both …
Research explores real algebraic realization of round fold maps of codimension -1.
problem Real algebraic realization of round fold maps of codimension -1.
method Generalizes canonical projections of unit spheres to round fold maps and discusses their real algebraic realization.
result Developed new studies in real algebraic geometry focusing on round fold maps of codimension -1.
The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.
problem Deriving Liouville theorems for generalized maps on Riemannian manifolds.
method Using conservation laws and monotonicity formulas, the paper derives Liouville theorems for different types of maps under various conditions.
result The paper establishes Liouville theorems for several types of generalized maps, including φ-F harmonic maps, φ-F symphonic maps, and φ-F-V-harmonic maps. This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Paper defines and studies Clairaut warped product Riemannian maps.
problem Understanding the geometry of specific Riemannian maps.
method Identify geodesic conditions, derive conditions for Clairaut maps, and calculate curvature.
result Found conditions for a warped product Riemannian map to be Clairaut.
The paper explores unique continuation properties for polyharmonic maps between Riemannian manifolds.
problem Investigating unique continuation principles for polyharmonic maps.
method Analyzing critical points of higher order functionals to prove extensions of known results in harmonic and biharmonic cases.
result Proving extensions of unique continuation principles for k-harmonic maps.
The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping…
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…
The paper constructs gluing maps for harmonic maps between Riemannian manifolds.
problem Constructing harmonic maps between Riemannian manifolds.
method Gluing construction of extended harmonic maps.
result Construction of gluing maps for harmonic maps under specific conditions.
Characterizes a general range decreasing group homomorphism.
problem Understanding range decreasing group homomorphisms in the entire mapping group.
method Characterization of a general range decreasing group homomorphism.
result Computes a particular class of homomorphisms and identifies all range decreasing group homomorphisms on specific mapping groups.
The paper examines HM-tensional and HS-tensional maps between Riemannian manifolds.
problem Analyzing tension fields of maps between Riemannian manifolds.
method Investigating harmonic maps and harmonic sections as tension fields.
result Characterization and properties of HM-tensional and HS-tensional maps. The paper proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
We introduce slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds as a generalization of slant immersions, invariant Riemannian maps and anti-invariant Riemannian maps. We give examples, obtain characterizations and investigate the harmonicity of such maps. We also obtain necessary and sufficie…
Analyzes harmonic and biharmonic maps from gradient Ricci solitons.
problem Characterizing maps from gradient Ricci solitons.
method Derives conditions for maps to be constant or harmonic.
result Biharmonic maps of finite energy from the two-dimensional cigar soliton are harmonic.
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂b- and ∂b-harmonic maps. result Generalizes Siu's holomorphicity result to ∂b- and ∂b-harmonic maps. In this paper, we show that one can interrelate pluriharmonic maps with para-pluriharmonic maps by means of the loop group method. As an appendix, we give examples for the interrelation between pluriharmonic maps and para-pluriharmonic maps. Moreover, we investigate the relation among CMC-surfaces by use of such maps.
The article explores constructing biharmonic and conformal biharmonic maps to spheres.
problem Constructing biharmonic and conformal biharmonic maps to spheres.
method Geometric algorithm to render harmonic maps biharmonic or conformally biharmonic.
result Explicit critical points for conformal-biharmonic maps between spheres are found.
Harmonic map flow preserves almost-holomorphic maps without singularities.
problem Preserving almost-holomorphic maps without singularities.
method Harmonic map flow applied to almost-holomorphic maps.
result No singularities or necks appear in the limit at singular times.
Method computes harmonic and conformal maps from point clouds.
problem Computing maps from irregular point cloud data.
method Meshless method using cubic lattice approximations.
result Harmonic and conformal maps computed accurately.
New theorem proves convergence of various discrete conformal structures to conformal maps.
problem Proving convergence of discrete conformal structures to conformal maps.
method General theorem using piecewise linear discrete conformal mappings and Riemannian barycentric coordinates.
result Discrete conformal mappings converge to conformal maps under certain conditions.
Constructs harmonic maps between special geometric shapes.
problem Creating harmonic maps between specific types of geometric shapes.
method Equivariant harmonic maps constructed between cohomogeneity one manifolds.
result Developed a method to construct harmonic maps.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.