Characterizes knot-theoretic flocks up to 64 elements.
problem Classifying ternary quasigroups for knot theory.
method Group action on flock colorings to improve knot-theoretic invariant.
result Enumerated and characterized knot-theoretic flocks up to 64 elements.
3D transverse links created from complex surfaces and spheres.
problem Creating 3D transverse links from complex surfaces and spheres.
method Various techniques, including constructions of quasipositive knots and links.
result Many 3-manifolds realized as transverse intersections of complex surfaces and strictly pseudoconvex 5-spheres.
We study the problem of finding the minimal (maximal) genus for a surface where a given four-valent graph with fixed opposite edge structure can be embedded into. We find several partial relations and give new reformulations in combinatorial and knot theoretic languages.
Quantum physics model uses knot theory for fragile topology.
problem Modeling quantum physics' fragile topology.
method Knot theoretic algorithm.
result Quantum physics' fragile topology modeled.
Mathematical pipeline identifies structural homology of knotted proteins.
problem Quantification and classification of protein structures, especially knotted proteins, require noise-free and complete data.
method Developed a geometric framework using persistent homology to analyze protein structures.
result Persistent homology accurately represents structural homology of knotted proteins and identifies geometric features of protein entanglement.
Ternary groups from knot theory help classify curves.
problem Characterizing ternary groups with knot theory axioms.
method Using semi-commutativity and Reidemeister moves.
result Constructs a curve invariant under Reidemeister moves.
Paper defines untangling number to measure entanglement complexity in 3-periodic networks.
problem Measuring the complexity of entanglement in 3-periodic networks.
method Defining ground states through knot-theoretic crossing diagrams and measuring untangling number.
result Introduced untangling number as a measure of entanglement complexity.
Shadow biquandles and local biquandles have similar homology and invariants.
problem Comparing homology and invariants of shadow biquandles and local biquandles.
method Defined local biquandle structure on a shadow biquandle and showed isomorphic (co)homology groups and invariant equivalence.
result Homology and invariants of shadow biquandles and local biquandles are equivalent.
Solves asymptotic An-realization problem for curves.
problem Realization problem for plane curves.
method Asymptotic analysis of smooth An-realization. result Determines cobordism distance between specific knot types.
New knot theory module shows torsion-ness in number theory.
problem Torsion-ness of Selmer modules in Galois representations.
method Introducing adjoint homological Selmer module for SL2-representations of knot groups. result Finitely generated torsion-ness of the new Selmer module.
We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple. Furtherm…
Explains how knots relate to 4D shapes.
problem Understanding 4D shapes through knot theory.
method Combines knot theory with 4D manifold topology.
result Connects 4D shapes to knot theory and other geometries.
We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic fixed point in a flow on ℜ3 originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or ``pleat,'' we may apply the theory of templates, or branched …
The paper classifies palettes of Dehn colorings for spatial graphs.
problem Classifying spatial graph diagrams using Dehn colorings.
method Examining vertex conditions and palettes for spatial graphs.
result Spatial graphs can be distinguished by the number of Dehn colorings with specific palettes.
The study confirms properties of specific knots and conjectures.
problem Verifying the Kervaire Conjecture for specific knots.
method Analyzing alternating Montesinos knots and pretzel knots.
result Alternating Montesinos knots with three tangles and pretzel knots of form P(p,q,r) have property Z. Alexander polynomial degree bounds twice a knot's topological slice genus.
problem Determining the topological slice genus of knots.
method Using Freedman's disc theorem and Alexander polynomial properties.
result The degree of the Alexander polynomial is an upper bound for twice the topological slice genus.
The Penrose-Kauffman polynomial connects knot theory to graph coloring.
problem Understanding the Penrose-Kauffman polynomial for cubic graphs.
method Using knot theory, the polynomial is shown equivalent to 3-coloring link diagrams.
result The Four Color Theorem is linked to 3-coloring link diagrams.
Study complexities of 3-manifolds using triangulations, Heegaard splittings, and surgeries.
problem Estimating complexities of 3-manifolds from different presentations.
method Showed linear inequalities between complexities defined by triangulations, Heegaard splittings, and surgeries.
result Linear inequalities are asymptotically optimal and used to estimate Cheeger-Gromov L2 ρ-invariants. Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. …
A graph G is called "minimalizable" if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called "strongly minimalizable". In this article, it is explained how minimalizability of a gr…
The article constructs a new map for fluid dynamics and reinterprets linking numbers.
problem Understanding higher order linking numbers in fluid dynamics.
method Hydrodynamical homotopy co-momentum map and multisymplectic interpretation.
result Reinterpretation of higher order linking numbers as conserved quantities.
Formula for interior polynomial of bipartite graphs derived from knot theory.
problem Deriving a formula for the interior polynomial of bipartite graphs.
method Applied knot theory, Ehrhart reciprocity, flyping and mutation.
result Proved a mirroring formula for the interior polynomial of bipartite graphs.
Researchers found a q-series identity for a specific knot using sl3 representations.
problem Finding a q-series tail for sl3 colored Jones polynomials. method Explicit formulas for the tail of sl3 colored Jones polynomials for (2,2m)-torus links. result An identity of q-series connecting sl3 colored Jones polynomials and Ramanujan false theta function. This paper outlines an approach to the non-abelian theta functions of the SU(2)-Chern-Simons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier tran…
This paper computes knot invariants using w-knotted objects.
problem Computing knot invariants of w-knotted objects.
method Introduces mathematical and computational tools to solve equations in Aw spaces. result Carries out computations of knot invariants up to a certain degree.
This paper applies knot theory to modern yo-yo play.
problem No systematic knot-theoretic treatment of contemporary yo-yo play.
method Recalled fundamental knot theory results and developed a methodology for classifying string arrangements (mounts).
result Classified a range of mounts and identified corresponding yo-yo maneuvers.
A graph connects Specht and web bases; matrix is unipotent with vanishing entries.
problem Comparing two bases of irreducible representations of the symmetric group.
method Graph theory and combinatorial analysis to describe relations between bases and prove properties of the transition matrix.
result The transition matrix between Specht and web bases is unipotent with additional vanishing entries.
This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C. The techniques currently available in the literature are either too theoretical, applying to only a small number of knots, or are designed to only deal with a specific knot. The thesis buil…
Investigates ropelength of complex knots and links.
problem Establishing ropelength for knots and links beyond tested crossing numbers.
method Investigated torus knots up to 1023 crossings, satellite knots up to 42 crossings, and used a model of repeated Hopf links.
result Found power-law scaling in T(p,p+1) and derived formulae for predicting crossing-ropelength relationships.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding the relationship between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving results for specific examples.
result Establishes compatibility for Poisson, contact, and locally conformally symplectic structures.
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
Defines compatibility between Riemannian and Jacobi structures.
problem Understanding compatibility between Riemannian and Jacobi structures.
method Introducing a notion of compatibility and proving it for specific structures.
result Fundamental examples of Jacobi structures lead to known structures like Riemann-Poisson, Kenmotsu, and locally conformally Kähler.
Extending Jacobi and Riemannian compatibility to Lie algebroids.
problem Generalizing compatibility between Jacobi and Riemannian structures.
method Generalizing previous work on fundamental examples to Lie algebroids.
result Compatibility results for Lie algebroids.
Study projective and direct limits of Banach structures with connections to G-structures.
problem Understanding connections between Banach structures and G-structures. method Endow projective and direct limits with Fréchet or convenient structures and study connections.
result Illustrated examples demonstrate the study of projective and direct limits.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
Study on G2∗ structures and almost para-contact structures in 7D.
problem Understanding the relation between G2∗ structures and almost para-contact structures. method Calculating projections using properties of G2∗ structures. result Determined the class of almost para-contact structures induced by G2∗ structures. Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of S2-family of generalized complex structures and study of twistor spaces. result Existence of generalized hypercomplex structures on 4n-dimensional tori with non-maximal types. Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
Study G2 structures on manifolds to find specific almost contact metric structures.
problem Understanding G2 structures and their induced almost contact metric structures. method Analyzing 2-fold vector cross products and their effects on manifolds.
result Identified possible classes of induced almost contact metric structures.
This article connects PCS structures to parabolic contact structures.
problem Connecting PCS structures to parabolic contact structures.
method Developed a parabolic version of contactification to show any PCS-structure can be locally realized.
result Any PCS-structure can be locally realized uniquely up to isomorphism in terms of parabolic contact structures.
Bihamiltonian structures lead to tau structures for integrable hierarchies.
problem Classifying deformations of bihamiltonian structures.
method Starting from flat exact semisimple bihamiltonian structures, we derive Frobenius manifolds and tau structures.
result Deformations of the principal hierarchy with tau structures are classified.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
Study on submanifolds in metallic structures with new results and structures.
problem Investigating submanifolds in metallic structures.
method Analyzing hypersurfaces and products spaces, defining new structures, and expressing fundamental theorems.
result New fundamental theorems for submanifolds in metallic structures.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Study GL(2)-structures on manifolds leading to complex structures.
problem Understanding GL(2)-structures and their relation to complex structures. method Explored GL(2)-structures on differential manifolds, proving their relation to almost-complex structures and providing a canonical connection. result Established a twistor-like construction for GL(2)-geometry.