Invariants for colored links found, with topological protection of certain knots.
problem Finding invariants for colored links and understanding their topological stability.
method Equivariant bordism groups of Conner and Floyd, topological surgeries.
result Topological protection of certain tricolored knots, with specific transformations.
A combinatorial presentation of closed orientable 3-manifolds as bi-tricolored links is given together with two versions of a calculus via moves to manipulate bi-tricolored links without changing the represented manifold. That is, we provide a finite set of moves sufficient to relate any two manifestations of the same …
Enhanced Kauffman bracket invariant in polynomial rings.
problem Improving Kauffman's bracket invariant for tricolorable links.
method Generalized Kauffman's bracket to polynomial rings and introduced a new invariant for lower crossing bounds.
result New invariant gives lower bounds on crossings for tricolorable links.
This paper shows all elements in the 3-colorable subgroup of Thompson's group give 3-colorable links.
problem Exploring the relationship between elements in the 3-colorable subgroup of Thompson's group and 3-colorable links.
method Defined the 3-colorable subgroup and used Jones's method to construct knots and links from elements of Thompson's group.
result All elements in the 3-colorable subgroup give 3-colorable links.
Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.
problem Investigate quotients of Gordian and H(2)-Gordian graphs under knot invariants.
method Defined equivalence relations by knot invariants (det, Jones span, tricolorability) and showed quotient graphs are Gromov hyperbolic.
result Quotients of H(2)-Gordian graph of links modulo span of Jones polynomial is isomorphic to complete graph.
Extends link colorings to modules over Laurent polynomial rings, showing isomorphisms and dimensions.
problem Extending link colorings to modules over Laurent polynomial rings.
method Using Alexander quandles and modules over Laurent polynomial rings, showing isomorphisms and dimensions.
result Dimension of colorings as vector spaces over fields is determined by ring homomorphisms.
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.
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.
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.
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.
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.
Link colorings linked to Goeritz matrix.
problem Understanding link colorings and their relation to the Goeritz matrix.
method Exploring the relationship between link colorings and the Goeritz matrix.
result Established a connection between link colorings and the Goeritz matrix.
Link Floer homology detects split links.
problem Detecting split links in link Floer homology.
method Module structure on link Floer homology and sutured Floer homology results.
result Link Floer homology detects split links.
Jones polynomial gaps removed for quasi-alternating links.
problem Proving Jones polynomial gaps for quasi-alternating links.
method Proving Jones polynomial properties through link twisting and conjecturing for prime quasi-alternating links.
result Jones polynomial gaps removed for certain quasi-alternating links.
This paper classifies virtual string links up to cobordism using group theory.
problem Classifying virtual string links up to cobordism.
method Using group theory, specifically the group Zn(n−1). result Virtual string links up to cobordism are classified by elements of Zn(n−1). New link types derived from old links.
problem Understanding the structure of modular links.
method Analyzing the complements of arithmetic modular links and comparing them to augmented chainlinks.
result Arithmetic modular links are homeomorphic to augmented chainlinks.
New invariant refines Milnor's triple linking number, revealing more information for complex links.
problem Indeterminacy of Milnor's triple linking number in complex link configurations.
method Introduced a new invariant called the total triple linking number, refining Milnor's original.
result The total triple linking number is non-trivial for every (n≥6)-component link, providing more information than classical triple linking numbers. Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…
Extends Benard-Conway invariant to all two-component links.
problem Counting irreducible SU(2) representations for two-component links.
method Counting irreducible SU(2) representations with fixed meridional traces.
result Invariant equals symmetrized multivariable link signature for (2, 2n)-torus links.
The paper extends CF-moves to classify virtual links of any number of components.
problem Classifying virtual links using CF-moves.
method Extending CF-moves to classify virtual links of arbitrary number of components using the virtual linking number and invariants.
result Classification of 3-component even virtual links up to CF-moves.
The paper confirms a conjecture and extends arrow polynomial to twisted links.
problem Extending classical invariants to virtual and twisted links.
method Checkerboard framings, cut points, and normalized arrow polynomial.
result The normalized arrow polynomial is an invariant for twisted links.
New obstructions show links with vanishing Milnor invariants may not be concordant to homology boundary links.
problem Understanding links with vanishing Milnor invariants and their concordance properties.
method Developing new obstructions and examples within the solvable filtration framework.
result Existence of links with vanishing Milnor invariants that are not concordant to homology boundary links.
New examples show limits of physical link isotopies.
problem Limits of physical link isotopies in thick and thickly embedded links.
method Construction of specific examples of thick and thickly embedded links.
result Explicit examples of links that cannot be split through thick homotopies.
Researchers calculate link Floer homology for 2-component L-space links.
problem Computing link Floer homology for specific L-space links.
method Using the h-function of the filtered chain complex determined by Alexander polynomials. result Explicit determination of Thurston polytope and norm for 2-component L-space links.