The study of semi-regular biperiodic alternating links and their geometric properties.
problem Understanding the geometric properties of biperiodic alternating links.
method Analyzing semi-regular links in the thickened torus, determining volumes, and relating commensurability and arithmeticity to Euclidean tilings.
result There exist infinitely many pairwise incommensurable semi-regular links with the same invariant trace field.
Research determines criteria for semi-regular tilings in hyperbolic space.
problem Finding combinatorial criteria for semi-regular tilings in hyperbolic geometry.
method Combinatorial analysis of vertex-types and geodesic polygons.
result Determined criteria for existence and uniqueness of semi-regular tilings.
In this article the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Rie…
A CNN on semi-regular meshes classifies brain diseases from MRI scans.
problem Classifying brain diseases from MRI scans.
method Developed a vertex-based graph CNN for semi-regular triangulated meshes.
result Vertex-based graph CNN outperformed spectral graph CNN in classifying MCI and AD.
New constructions of Sasakian and K-contact structures on Smale-Barden manifolds.
problem Deciding when a Smale-Barden manifold admits a Sasakian or K-contact structure.
method Developing quasi-regular Seifert fibrations and applying them to constructions.
result Determined all Smale-Barden manifolds admitting null Sasakian structures and provided counterexamples to conjectures.
It is shown that the Schwarzschild spacetime can be extended so that the metric becomes analytic at the singularity. The singularity continues to exist, but it is made degenerate and smooth, and the infinities are removed by an appropriate choice of coordinates. A family of analytic extensions is found, and one of thes…
The paper studies affine connections on singular warped products and their curvature.
problem Analyzing affine connections on singular warped products.
method Introducing semi-symmetric metric and non-metric Koszul forms, and expressing their curvature in terms of factor manifolds.
result Generalized results for singular multiply warped products.
New 5D manifold found without certain Sasakian structure.
problem Finding a 5D manifold without specific Sasakian structure.
method Constructing symplectic 4-manifold and proving Betti number bound.
result First example of 5D manifold with K-contact but no semi-regular Sasakian structure.
On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related o…
New methods distinguish K-contact from Sasakian structures on 5-manifolds.
problem Distinguishing K-contact from Sasakian structures on 5-manifolds.
method Develop methods using Kollár's obstructions to find subtle differences.
result First example of a K-contact 5-manifold without semi-regular Sasakian structures.
In this paper, we study deformations of compact holomorphic Poisson submanifolds which extend Kodaira's series of papers on semi-regularity (deformations of compact complex submanifolds of codimension 1), deformations of compact complex submanifolds of arbitrary codimensions, and stability of compact complex submanifol…
We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair (p,q), 2<p<q, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular pq-fold covering…
This work classifies Smale-Barden manifolds with Sasakian structures.
problem Deciding when Smale-Barden manifolds admit Sasakian structures.
method Classifying manifolds by their homology and Barden invariant, and providing methods to realize specific cases.
result Realized all manifolds with specified homology and Barden invariant, contributing to the existence of Sasakian structures on rational homology spheres.
New semi-equivelar map found on sphere, not vertex-transitive.
problem Characterizing semi-equivelar maps on sphere.
method Combinatorial proofs and vertex-transitivity analysis.
result Exactly one semi-equivelar map on sphere is not vertex-transitive.
The paper calculates actions of string link operations for 4- and 5-component links.
problem Classifying link-homotopy classes of string links.
method Explicit calculation of actions of partial conjugations and conjugations for 4- and 5-component string links.
result Presentations of link-homotopy classes for 4- and 5-component links.
Study proves chainmail links are L-space links.
problem Understanding L-space links in chainmail links.
method Inspired by alternating links, uses flat augmentation.
result Proves alternating chainmail links are L-space links.
New findings on T-links derived from torus links.
problem Difficulty in determining geometric type of T-links.
method Examined T-links obtained by full twists along torus links.
result T-links obtained by full twists are not torus links.
Paper discusses when virtual links are equivalent as twisted links.
problem Determining equivalence of virtual links as twisted links.
method Using stable equivalence classes of links in oriented thickenings of surfaces.
result Necessary and sufficient condition for virtual links to be equivalent as twisted links.
Abstract lists known link diagrams for all principal congruence link complements.
problem Identifying all known link diagrams for principal congruence link complements.
method Compilation of known link diagrams.
result Compilation of known link diagrams for principal congruence link complements.
Classifies colored links and spatial graphs up to colored link-homotopy.
problem Classifying colored links and spatial graphs up to colored link-homotopy.
method Using Habegger-Lin theory for colored string links, and extending to colored links and spatial graphs.
result Classification of colored links and spatial graphs up to colored link-homotopy.
Paper finds linking numbers for Montesinos links using a simple algorithm.
problem Calculating linking numbers for Montesinos links.
method Simple proof and numerical algorithm for rational links, extending to Montesinos links.
result Linking numbers found for any two components in Montesinos links.
We define and prove properties of link lattice complexes for plumbed links.
problem Understanding the homology and formality of plumbed L-space links.
method Define and analyze link lattice complexes, proving homotopy equivalence and formality properties.
result Link lattice complexes are homotopy equivalent to link Floer complexes for plumbed links.
Study of Lorenz links and T-links, showing equivalence and unique presentations.
problem Understanding equivalence and uniqueness of T-link presentations for Lorenz links.
method Minimal-braid approach, V-link braids, T-link parameters, geometric type criteria.
result Explicit correspondence between V-links and T-links, criteria for equivalence and uniqueness.
New links homotopic but not isotopic to a given link.
problem Constructing links that are homotopic but not isotopic.
method Constructing a family of links with specific properties.
result Existence of links homotopic but not isotopic to a given link.
Paper proves Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
problem Proving relations between Reshetikhin-Turaev and re-normalized link invariants for links.
method Analyzing higher order terms of re-normalized link invariants for plumbed links.
result Reshetikhin-Turaev link invariants appear in higher order terms of re-normalized link invariants for plumbed links.
Method converts virtual link diagrams to normal form, preserving equivalence.
problem Normalizing virtual link diagrams to study their properties.
method Method converts virtual link diagrams to normal form.
result Normal virtual link diagrams from equivalent diagrams are equivalent.
New method classifies 4-component link homotopy using claspers.
problem Classifying link-homotopy classes of 4-component links.
method Using Habiro's clasper theory to modify Levine's algebraic computations.
result More symmetrical and schematic classification of 4-component link homotopy.
Extends positive and almost positive links to successively almost positive ones.
problem Extending properties of positive and almost positive diagrams and links.
method Introducing successively almost positive diagrams and links, and analyzing their properties.
result Improves known results of positive and almost positive links.
Study of knots and links in 2-complexes, defining linking numbers and polynomials.
problem Understanding knots and links in 2-dimensional complexes.
method Definition of linking numbers and Kauffman-type bracket polynomials for links in 2-complexes.
result Established relationships between 2-complexes and knots/links in 3-manifolds.
Study Milnor invariants for covering links to distinguish certain links.
problem Distinguish links using Milnor invariants for covering links.
method Generalize Hartley and Murasugi's covering linkage invariants and use cobordism invariants.
result First non-vanishing Milnor invariants of a Brunnian link modulo 2 equals a sum of linking numbers of covering links.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
We generalized the periodic links to \emph{transitive} links in a 3-manifold M. We find a complete classification theorem of transitive links in a 3-dimensional sphere R3. We study these links from several different aspects including polynomial invariants using the relation between link polynomials of…
Characterizes a subset of links using quasipositive and homogeneous properties.
problem Understanding the properties of T-positive links.
method Characterization through strongly quasipositive and T-homogeneous braids.
result T-positive links are precisely the strongly quasipositive links that are closures of T-homogeneous braids.
New link invariants from colored link polynomial.
problem Constructing stronger link invariants.
method Using skein invariant of colored links.
result Invariants stronger than HOMFLYPT polynomial.
Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.
Study maps on link Floer homology for unoriented link cobordisms.
problem Defining maps on link Floer homology induced by unoriented link cobordisms.
method Introduced disoriented link cobordism to track critical points, constructed a map on unoriented link Floer homology.
result Comparison with existing constructions for unoriented band moves.
New link invariants from diagram colorings match link widths.
problem Defining link widths via diagram colorings.
method Colorings of link diagrams to define invariants and prove their equivalence to link widths.
result Invariants of link widths calculated algorithmically.
Innovates a three-component link homotopy invariant.
problem Classifying three-component link maps up to homotopy.
method Developed tools and invariants for distinguishing three-component link maps.
result Found three-component link maps that are not homotopic.
Generalizes surface-link normal forms to immersed links.
problem Normal forms for surface-links and their immersions.
method Introduces ribbon-clasp surface-links and proves normal forms.
result Any surface-link can be described in a symmetric normal form.
Enhanced Alexander module detects linking numbers in links.
problem Detecting linking numbers in links using Alexander modules.
method Defining and singling out meridians and longitudes in reduced Alexander modules.
result The enhanced Alexander module determines all linking numbers.
String links classified up to 2n-moves and link-homotopy.
problem Classifying string links up to specific moves and link-homotopy.
method Using Milnor link-homotopy invariants and 2n-moves. result Equivalence classes of string links form a finite group.
The paper uses Heegaard Floer homology to find lower bounds on link genera.
problem Finding lower bounds on the 4-genus of links with vanishing pairwise linking numbers.
method Using the h-function from Heegaard Floer homology to describe genera of surfaces bounded by link components.
result The h-function provides lower bounds for the 4-genus of links, and these bounds are sharp for some L-space links.
Study shows hyperbolic links are not common among prime links.
problem Understanding the prevalence of hyperbolic links among prime links.
method Analyzing satellite and prime non-split links of varying complexity.
result The proportion of hyperbolic links among all prime non-split links does not approach 1 as complexity increases.
The theory of signature invariants of links in rational homology spheres is applied to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, an explicit formula is derived to compute signature invariants of their covering links. Using the formula, we produce fused bou…
The paper classifies links up to link-homotopy using claspers.
problem Classifying links up to link-homotopy.
method Using Habiro's clasper calculus, defining a linear representation of the homotopy braid group, and providing a geometric proof.
result Geometric proof of Levine's classification of 4-component links and further classification of 5-component links in the algebraically split case.
New proof of link classification using map invariants.
problem Classifying two-component links in S3 and string links. method Invariants of link maps S2⊔S2oS4 and their variations. result New proof of Nakanishi-Ohyama classification for two-component links.
New methods show hyperbolicity of Brunnian links.
problem Determining hyperbolicity of Brunnian links.
method Geometric/combinatorial topology techniques applied to links with at least one unknotted component.
result Determine hyperbolicity for all Brunnian links.
The paper examines when links can have homeomorphic C-complexes.
problem When do two links have homeomorphic C-complexes?
method The paper investigates the conditions under which two links admit homeomorphic C-complexes, focusing on pairwise and triple linking numbers.
result Pairwise linking number provides a complete obstruction for 2-component links, while Milnor's triple linking number provides a complete obstruction for 3 or more components with zero pairwise linking number.