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.
The paper studies isotopies of surfaces in 4-manifolds using banded unlink diagrams.
problem Isotopies of surfaces in 4-manifolds.
method Complete set of moves relating banded unlink diagrams of isotopic surfaces.
result Bridge trisections of isotopic surfaces in a trisected 4-manifold are related by a sequence of perturbations and deperturbations.
New rational band moves simplify knot classification.
problem Classifying knots using rational moves.
method Introduced oriented rational band moves and proved their effectiveness.
result Knots that can be unlinkified by rational moves are rationally slice.
Generates minimal hard surface-link diagrams up to ten crossings.
problem Finding minimal diagrams for surface-links.
method Describes a method for generating minimal hard prime surface-link diagrams.
result Generates all minimal hard prime surface-unknot and surface-unlink diagrams up to ten crossings.
Study shows torsion order bounds band-unlinking number for knot cobordisms.
problem Understanding bounds on knot cobordisms using torsion orders.
method Established an inequality involving local maxima, genus, and torsion orders of Khovanov homology.
result Torsion order gives a lower bound for the band-unlinking number.
Simplified plat diagrams for unlink without stabilization.
problem Equivalence of plats without stabilization for unlink.
method Introducing pocket and flip moves to simplify plats.
result Simplified plat diagrams for unlink using pocket and flip moves.
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.
Study categorizes knots and links as rigid or shaky based on Reidemeister moves.
problem Classifying knots and links as rigid or shaky based on adaptability to Reidemeister moves.
method Categorization of hard diagrams as rigid or shaky, investigation of rigid and shaky hard diagrams for specific knots and links.
result Every link has a rigid hard diagram, and there is an upper limit for the number of crossings in such diagrams.
Enhances Hantzsche's theorem for 3-manifolds in 4D.
problem Embedding 3-manifolds in 4-dimensional space.
method Using Heegaard diagrams and a property called doubly unlinked (DU).
result Enhanced Hantzsche's embedding obstruction.
Study sharpens unlinking number bounds for special alternating links.
problem Determining the exact unlinking number for special alternating links.
method Analyzes links in the 3-sphere, focusing on special alternating links and their crossing changes.
result Sharp lower bounds for unlinking number realized by crossing changes in alternating diagrams.
Proves Yoshikawa eighth move's independence and finds minimal band moves.
problem Proving the independence of Yoshikawa eighth move and finding minimal generating set of band moves.
method Defining twisted diagrams and mirror cut surfaces, using surface-link groups.
result Proves independence of Yoshikawa eighth move and finds minimal generating set of band moves.
New invariant shows fifth move is unique and extends biquandle theory for surface-links.
problem Reproduce Yoshikawa's fifth move using other moves and isotopies.
method Develops a semi-invariant and biquandle-based invariant for surface-links.
result Yoshikawa's fifth move is unique and cannot be reproduced by other moves.
New bounds on knot complexity defined via band number and surface diagrams.
problem Understanding the complexity of knots and their generalizations.
method Defining and analyzing band number, using broken surface diagrams and 3-manifold embeddings.
result Band number is an upper bound on double slice genus, a knot complexity measure.
Knot theory problems are proven NP-hard or NP-complete.
problem Problems in knot theory, like diagram transformations and unlinking, are computationally hard.
method Proved NP-hard or NP-complete using Reidemeister moves and sublink analysis.
result Certain knot theory problems are computationally intractable.
Delta-unlinking number measures how to unlink algebraically split links.
problem Measuring unlinking complexity of algebraically split links.
method Defining delta-unlinking number as minimum delta-moves to unlink, proving bounds and calculating specific values.
result Precise delta-unlinking numbers for algebraically split prime links up to 9 crossings, and 4-genus values for most.
New method for quandle presentations of surface knots in 4-manifolds.
problem Computing fundamental quandle presentations for surface knots in arbitrary 4-manifolds.
method Wirtinger type presentation of the fundamental quandle for surface links in 4-manifolds.
result Infinitely many pairwise non-local surface knots with specific bridge numbers in certain 4-manifolds.
Study finds new knot distances and chirally cosmetic bands using grid diagrams.
problem Computing distances and identifying chirally cosmetic bands between knots.
method Using grid diagrams to explore non-coherent band attachments between knots.
result Discovery of 33 new H(2)-distance one pairs for knots up to 8 crossings.
The abstract extends Reidemeister theorem to 3-manifolds using diagrams of links and bands.
problem Extending Reidemeister theorem to 3-manifolds.
method Defines diagrams for links and bands on spines and flow-spines of 3-manifolds and describes combinatorial moves.
result Reproves and extends Brand et al. result on 3-manifolds.
New tangles found for 21 ribbon knots with 10 or fewer crossings.
problem Characterizing ribbon knots through tangle forms.
method For each of 21 ribbon knots, found a 4-strand tangle that satisfies specific closure properties.
result Each of 21 ribbon knots can be represented by a tangle that meets the specified closure conditions.
In the classical knot theory there is a well-known notion of descending diagram. From an arbitrary diagram one can easily obtain, by some crossing changes, a descending diagram which is a diagram of the unknot or unlink. In this paper the notion of descending diagram for knots and links in the real space is extended to…
In the previous paper, we considered a link diagram invariant of Hass and Nowik type using regular smoothing and unknotting number, to estimate the number of Reidemeister moves needed for unlinking. In this paper, we introduce a new link diagram invariant using irregular smoothing, and give an example of a knot diagram…
Permutations linked to knots and links, with unknots counted by Schröder numbers.
problem Understanding permutations as knots and links.
method Using grid diagrams and Bennequin's inequality.
result Permutations corresponding to unknots and links are counted by Schröder numbers.
Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application, we show that a certain infinite sequence of diagrams of the trivial two-component link need quadratic number of Reidemeister moves for being unknotted with resp…
A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary cir…
Axel Seeliger tabulated ribbon knots and found non-symmetric examples.
problem Identifying non-symmetric ribbon knots.
method Tabulated symmetric union presentations and band diagrams for ribbon knots.
result Found non-symmetric ribbon knots with 11 and 12 crossings.
A new invariant for links in lens spaces with advantages over traditional methods.
problem Invariants for links in lens spaces.
method Constructing a virtual quandle for links in lens spaces L(p,1). result The virtual quandle is an essential invariant and its presentation can be easily written from a link's band diagram.
Knot invariants and quiver stability linked through full twists.
problem Relating knot invariants to quiver stability under twists.
method Using HOMFLY-PT skein relations and linking/unlinking operations on symmetric quivers.
result Full twists on knots correspond to unlinking or linking of augmented symmetric quivers, confirming stable growth in both.
Flip symmetry on knot diagrams affects Khovanov homology.
problem Understanding the flip map on Khovanov homology.
method Analyzing the behavior of the flip map on unlinks and using it to determine the involution.
result The flip map is the identity map over \(\mathbb{F}_2\), confirming a conjecture.
Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rect…
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 introduce a matrix representation of a chord on a tangle which leads us to representing tangle chord diagrams as stacks of matrices that we call books. We show that band sum moves, Reidemeister moves as well as orientation changes are implemented on \widetilde{Z}_f - a framed link invariant constructed from the Kont…
Computing unlinking number is usually very difficult and complex problem, therefore we define BJ-unlinking number and recall Bernhard-Jablan conjecture stating that the classical unknotting/unlinking number is equal to the BJ-unlinking number. We compute BJ-unlinking number for various families of knots and links for w…
The paper introduces triple grid diagrams to construct Lagrangian surfaces in complex projective space.
problem Constructing Lagrangian surfaces in complex projective space.
method Defining and analyzing triple grid diagrams to determine Lagrangian caps and surfaces.
result Triple grid diagrams can determine closed Lagrangian surfaces in CP2 under certain conditions. Paper finds first examples of unlinked knots that can't be separated.
problem Separating knots without changing their length and thickness.
method Constructs infinite families of 2-component gordian unlinks and n-component links for n≥2. result Found infinite families of 2-component gordian unlinks that cannot be separated.
Classical and welded knot moves are classified and their interrelationships discussed.
problem Classical and welded knot moves and their interrelationships.
method Algebraic classification and topological interpretations of local moves.
result All local moves are unknotting operations for welded knots.
Floer homology bounds knot properties like bridge index and ribbon distance.
problem Bounding knot properties using Floer homology.
method Floer homology applied to knot cobordisms.
result Lower bounds on bridge index and ribbon distance.
Study unlinking numbers of 10-crossing links using link invariants.
problem Investigate unlinking numbers of 10-crossing links.
method Use various link invariants and explore their behavior with crossing changes.
result Find the unlinking numbers of all but 2 of the 287 prime, non-split links with crossing number 10.
Develops robust persistence diagrams using kernel methods.
problem Persistence diagrams are sensitive to data perturbations.
method Constructs robust persistence diagrams from superlevel filtrations of robust density estimators using reproducing kernels.
result Robust persistence diagrams are consistent estimators in bottleneck distance.
The paper studies unlinking links with minimal crossing changes.
problem Unlinking links with minimal changes when crossings between components are fixed.
method Examined minimum number of crossing changes for links with linking number zero and unknotted components.
result Data and general results about asymmetry of unlinking between components for links with up to ten crossings.
New spanning 3-disks found for unlink in 4-sphere.
problem 2-component unlink in 4-sphere.
method Found infinitely many isotopy classes of Brunnian spanning 3-disks.
result Infinitely many Brunnian spanning 3-disks for 2-unlink in 4-sphere.
The paper classifies twisted torus links that are unlinks.
problem Identifying conditions for twisted torus links to be unlinks.
method Characterization of parameter families (p,q,r,s) for unlinking twisted torus links. result Complete characterization of parameter families for unlinking twisted torus links.
New math detects unlinks using HOMFLYPT homology.
problem Detecting unlinks in knot theory.
method Applied Rasmussen spectral sequence to HOMFLYPT homology over Z2. result HOMFLYPT homology over Z2 detects unlinks. Infinitely many splitting spheres found for unlinked 2-spheres in 4-space.
problem Existence of pairwise non-isotopic splitting spheres for unlinked 2-spheres in 4-space.
method Analytical proof showing non-isotopic spheres.
result Infinitely many non-isotopic splitting spheres found.
We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follows from topological Yoshikawa moves. …
The paper explores methods to construct knots with diffeomorphic 0-traces.
problem Constructing knots with diffeomorphic 0-traces. method Survey of Gompf-Miyazaki's dualizable pattern, Abe-Jong-Omae-Takeuchi's band presentation, and RGB-diagram.
result A sufficient condition for two knots obtained by Abe-Jong-Omae-Takeuchi's method to coincide.
New examples of gordian unlinks show different rope geometries.
problem Tackling the existence of non-trivial unknots with prescribed length and thickness.
method Provided the first examples of gordian unlinks with thickness in [1,2).
result Thinner normal tubes lead to different rope geometries.
Researchers find minimal-area metrics on surfaces with constraints.
problem Finding the conformal metric of least area on surfaces with length conditions.
method Numerical convex programs to solve the minimal-area problem.
result The metric is positively curved in some regions and flat in others.
Basket links are shown to be isotopic to T(2,n+1).
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to An matrix. result Basket links are isotopic to T(2,n+1).