New method classifies knots and links uniquely based on bridge number.
problem Classifying knots and links by crossing number is computationally infeasible.
method Using bridge number and geometric topology techniques, proving uniqueness of diagrams.
result Infinitely many knot and link diagrams have a unique simple m-bridge diagram under certain conditions. In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Each surface diagram co…
Minimal link diagrams are unique and have a fixed number of crossings.
problem Characterizing minimal link diagrams under Reidemeister moves.
method Analyzing minimal link diagrams through Reidemeister moves I and II.
result The uniqueness of minimal diagrams and their fixed number of crossings.
Defines knotting probability for spatial arcs.
problem Calculating the probability of knotting in spatial arcs.
method Projection of spatial arcs to planes, defining knotting probability for diagrams.
result Knotting probability defined for every oriented spatial arc.
Study of 2-knots via trisection diagrams, showing unique descriptions and operations.
problem Understanding and describing 2-knots via trisection diagrams.
method Use of doubly pointed trisection diagrams to describe and manipulate 2-knots.
result Unique descriptions of 2-knots via trisection diagrams, and operations on these diagrams.
There is a well-known way to describe a link diagram as a (signed) plane graph, called its Tait graph. This concept was recently extended, providing a way to associate a set of embedded graphs (or ribbon graphs) to a link diagram. While every plane graph arises as a Tait graph of a unique link diagram, not every embedd…
New method finds grid diagrams for many fibered knots.
problem Detecting fibered knots using grid diagrams.
method Developed an efficient method to identify grid diagrams with unique maximal Alexander grading states.
result Found suitable grid diagrams for 5385 of 5397 fibered prime knots with crossing number ≤ 13.
New method proves uniqueness of even plats diagrams.
problem Proving uniqueness of even plats diagrams.
method New arguments and restrictions on twist regions.
result Minimal width even plats diagrams are unique.
The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the warping matrix is one greater than the crossing number. We also discuss the linearly i…
Study surfaces in 4-manifolds using banded unlink diagrams.
problem Representing and manipulating immersed surfaces in 4-manifolds.
method Singular banded unlink diagrams and associated moves.
result Every immersed surface can be represented uniquely by a singular banded unlink diagram.
Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
A note on the uniqueness of differential characters and K-theory via homological algebra.
problem Existence and uniqueness of differential characters and differential K-theory.
method Observation and application of Rakesh Pawar's results in homological algebra.
result The hexagon diagram uniquely determines differential K-theory groups up to isomorphism.
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
Kuperberg [Algebr. Geom. Topol. 3 (2003) 587-591] has shown that a virtual knot corresponds (up to generalized Reidemeister moves) to a unique embedding in a thichened surface of minimal genus. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. Using this result a…
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
Algorithm constructs and classifies weaving diagrams using combinatorial methods.
problem Classifying unique weaving diagrams with over/under information.
method Systematic algorithm based on tiling and crossing matrices.
result Classification of periodic structures based on minimum crossings.
Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
Gauss codes help uniquely identify virtual doodles.
problem Identifying virtual doodles uniquely.
method Introduced left canonical Gauss codes.
result Oriented virtual doodles uniquely presented by left canonical Gauss codes.
Paper defines a polynomial invariant for surface-links using quantum A_2 invariant.
problem Defining a polynomial invariant for surface-links.
method Using the quantum A_2 invariant and Yoshikawa moves, a polynomial is defined for marked graph diagrams.
result The polynomial invariant is useful for studying ribbon 2-knots.
Paper proves uniqueness of bridge multisections for surfaces in 4-space.
problem Tackles the combinatorial description of surface links in 4-space.
method Develops a surgery operation called band surgery to prove uniqueness.
result Proves any n-valent graph is the spine of a bridge multisection for an unknotted surface. We show that totally real elliptic Lefschetz fibrations that admit a real section are classified by their "real loci" which is nothing but an S1-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a \emph{necklace diagram}. On the o…
New invariants distinguish symmetric union diagrams.
problem Distinguishing symmetric union diagrams.
method Introduced new invariants from topological spin models.
result Many effective invariants distinguish symmetric union diagrams representing the same link.
We refine a Le and Murakami uniqueness theorem for the Kontsevich Integral in order to specify the relationship between the two (possibly equal) main universal link invariants: the Kontsevich Integral and the perturbative expression of the Chern-Simons theory. As a corollary, we prove that the Altschuler and Freidel an…
This paper describes a polynomial invariant of virtual knots that is defined in terms of an integer labeling of the virtual knot diagram. This labeling is seen to derive from an essentially unique structure of affine flat biquandle for flat virtual diagrams. The invariant is discussed in detail with many examples,inclu…
A reinforcement learning framework evolves persistence diagrams on PD space for stochastic dynamics.
problem Stochastic modeling and evolution of persistence diagrams (PDs) for topological analysis.
method Reinforcement learning framework for PD space, evolving diagrams through topology-aware local edit operations.
result Established conditions for geometric ergodicity of Markov chains on PD space, yielding unique stationary laws.
Virtual braids are a combinatorial generalization of braids. We present abstract braids as equivalence classes of braid diagrams on a surface, joining two distinguished boundary components. They are identified up to isotopy, compatibility, stability and Reidemeister moves. We show that virtual braids are in a bijective…
Study of 3-handle attachments in 4D manifolds using Kirby calculus.
problem Inclusion of 3-handles in Kirby diagrams for 4D manifolds.
method Developed moves to extend Kirby calculus, established homological criterion.
result Identified geometric basis of 3-handle attachments, proved uniqueness theorem.
Let Πbe a link projection in S^2. John Conway and later Francis Bonahon and Larry Siebenmann undertook to split Π into canonical pieces. These pieces received different names: basic or polyhedral diagrams on one hand, rational, algebraic, bretzel, arborescent diagrams on the other hand. This paper proposes a thorough…
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…
A new type of knot energy is presented via real life experiments involving a thin resilient metallic tube. Knotted in different ways, the device mechanically acquires a uniquely determined (up to isometry) normal form at least when the original knot diagram has a small number of crossings, thus outperforming the famous…
Paper proves unique canonical form for certain highly twisted knots and links.
problem Classifying knots and links with specific plat projections.
method Analyzes 2m-plat projections with specific constraints on crossings and heights. result Unique canonical form for certain knots and links with plat projections.
The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.
problem Representing compact 4-manifolds using Kirby diagrams and graphs.
method Algorithmically constructing 5-colored graphs from Kirby diagrams to represent PL 4-manifolds.
result Upper bounds for gem-complexity and regular genus derived from Kirby diagrams.
String vertices proven in hyperbolic geometry, unique up to transformations.
problem Existence and uniqueness of string vertices in string field theory.
method Homological proof using hyperbolic metrics and geodesic boundaries.
result String vertices are sets of surfaces with systole greater than or equal to L. This paper introduces a homology theory for links in I-bundles over an orientable surface. The theory is unique in that the elements of the chain groups are surfaces instead of diagrams. It is then shown this theory yields the same results as the homology theory constructed by Asaeda, Przytycki and Sikora.
The study confirms a conjecture about polynomials related to symmetric spaces.
problem Understanding polynomials associated with isotropy orbits of symmetric spaces.
method Identified Reiswich's polynomials as special cases of Jacobi polynomials and proved their conjecture.
result The polynomials have pairwise different real roots in the interval [0,1].
Several new combinatorial descriptions of closed 4-manifolds have recently been introduced in the study of smooth maps from 4-manifolds to surfaces. These descriptions consist of simple closed curves in a closed, orientable surface and these curves appear as so called vanishing sets of corresponding maps. In the presen…
We prove every knotted surface can be isotoped to a bridge position in a 4-manifold.
problem Understanding knotted surfaces in 4-manifolds.
method Introducing generalized bridge trisections and shadow diagrams.
result Existence of exotic 4-manifolds with certain trisections.
We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the deco…
This paper classifies a specific weave type by their crossing number.
problem Classifying doubly periodic untwisted (p,q)-weaves.
method Classification by crossing number, introducing crossing matrix for equivalence.
result Classification of untwisted (p,q)-weaves by their crossing number.
Trivial links are unique up to number of link components, but they can be hard to recognize from arbitrary diagrams. We define a new measure of the complexity of a link embedding, the crumple, and show how this may be used to measure progress toward a trivial embedding. In conjunction with a modified form of arc presen…
Characterizes local tropicalizations of splice type surface singularities.
problem Understanding splice type surface singularities from a tropical geometry perspective.
method Characterization of local tropicalizations as cones over splice diagrams, using tropical methods.
result Characterizes local tropicalizations of splice type surface singularities as cones over associated splice diagrams.
Estimates BV functions from noisy data using Voronoi diagrams.
problem Estimating multivariate BV functions from scattered noisy data.
method Form Voronoi diagram, solve optimization problem with discrete TV regularization.
result Voronoigram is minimax rate optimal for BV functions.
New framework for 3D spatial topology enumeration and identification.
problem Efficient navigation through complex engineering system topologies.
method Mathematical spatial graph theory to represent, enumerate, and identify unique topological classes.
result Identification of distinctive 3D topological classes for engineering systems.
Algorithm converts curves on ribbon surfaces to contact surgery diagrams.
problem Legendrian realization of curves on ribbon surfaces.
method Explicit algorithm to Legendrian realize homologically nontrivial curves.
result Any two Legendrian realizations of the same curve are Legendrian isotopic.
New framework for sutured manifolds using handleslides and Heegaard invariants.
problem Naturality results for Floer homology theories in sutured manifolds.
method Definition of tight Heegaard invariants and their extensions to strong Heegaard invariants.
result Unique extensions of Heegaard invariants provide new frameworks for naturality results.
Legendrian knots in tight 3-sphere are uniquely determined by their exteriors.
problem Determining Legendrian knots in tight contact structures.
method Contactomorphism type of exterior, Thurston-Bennequin invariant formula.
result Not all Legendrian links in tight 3-sphere are uniquely determined by their exteriors.
We show that the only closed 4-manifolds admitting genus two trisections are S2×S2 and connected sums of S1×S3, CP2, and CP2 with two summands. Moreover, each of these manifolds admits a unique genus two trisection up to diffeomorphism. The proof relies heavily o…
The paper finds and visualizes unique geometric polyhedra and tori with few vertices.
problem Finding and visualizing geometric polyhedra and tori with specific vertex configurations.
method Using Schlegel diagrams and geometric realization in 3D and 4D space.
result Identifies and visualizes 12 triangulations of the 2-torus and 12 triangulations of the 2D projective plane.