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.
Defines new link-homotopy invariants using Milnor's higher order link invariants.
problem Link-homotopy invariants for link maps of multiple components.
method Uses Milnor's higher order link invariants and combinatorial theory of cut-diagrams.
result Provides practical algorithms to compute these invariants and detects families of examples.
The paper constructs new surface-link invariants using link invariants and applies them to the Kauffman bracket.
problem Constructing invariants for surface-links.
method Using invariants for knots and links in 3-space, the paper constructs ideal coset invariants for surface-links in 4-space.
result The paper introduces new invariants for surface-links, including the Kauffman bracket ideal coset invariant and a series of new invariants defined by skein relations.
We consider Milnor invariants for certain covering links as a generalization of covering linkage invariants formulated by R. Hartley and K. Murasugi. A set of Milnor invariants for covering links is a cobordism invariant of a link, and that this invariant can distinguish some links for which the ordinary Milnor invaria…
Combines combinatorial method to extend Milnor invariants to welded links.
problem Extending Milnor invariants to welded links.
method Combinatorial approach.
result Invariance of extended Milnor invariants for welded links.
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.
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.
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.
Constructs universal link invariants from intersections in configuration spaces.
problem Globalise topologically all coloured Jones polynomials and ADO polynomials.
method Defines new link invariants from graded intersections in configuration spaces.
result Recover all coloured Jones polynomials and ADO polynomials for links.
We study the eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha--Ko. We give a formula for the eta-invariant for boundary links. In several…
New Arf invariants for colored links determined by linking numbers.
problem Extending Arf invariant to colored links.
method Using generalized Seifert forms to construct quadratic forms and determining Arf invariant.
result New Arf invariants for colored links are determined by linking numbers.
New polynomial invariant distinguishes singular links.
problem Distinguishing singular links using existing invariants.
method Generalized quandle polynomial to singquandles and constructed a singular link invariant.
result New polynomial invariant distinguishes singular links with same counting invariant.
New invariants derived from Seifert graphs help distinguish alternating links.
problem Distinguishing alternating links from each other.
method Introducing new quantities derived from Seifert graphs of reduced alternating link diagrams and proving they are link invariants.
result These new invariants can easily distinguish many different alternating links, even large and complicated ones.
New algebraic invariants for singular links discovered.
problem Defining and proving invariants for singular links.
method Introduced tied links and tied braid monoid, proved decomposition, re-proved Alexander and Markov theorems.
result Introduced new five-variable polynomial invariants for tied singular links.
Invariants for virtual and twisted links using affine indices.
problem Computing invariants for virtual and twisted links.
method Using affine indices to define invariants for virtual and twisted links.
result Invariants for virtual and twisted links computed using affine indices.
New invariants for link concordance defined using Whitehead doubles.
problem Defining and analyzing new concordance invariants for links.
method Extending slice-torus invariants to links, proving properties, and using Whitehead doubles.
result New strong concordance invariants for links, independent of slice-torus link invariants.
Extends Milnor's invariants to knots and links in 3-manifolds.
problem Detecting higher-order linking phenomena in 3-manifolds.
method Study lower central quotients of link groups in 3-manifolds.
result Unified and generalized Milnor's invariants for 3-manifolds.
For an n-component link L, the Milnor's isotopy invariant is defined for each multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here m is called the length. Let r(I) denote the maximam number of times that any index appears. It is known that Milnor invariants with r=1 are link-homotopy invariant. N. Habegger and X. S.…
New invariants of links are constructed using the skein invariant polynomial of colored links defined by the author in [1]. These invariants are stronger than the homflypt polynomial.
New invariants defined for framed knots and links.
problem Defining invariants for framed knots and links.
method Introducing birack brackets and categorifying their multiset.
result Quiver-valued invariant defined for framed knots and links.
This paper establishes a correspondence between biquandle and quandle colorings for classical and surface links.
problem Finding refined invariants for classical and surface links using biquandles.
method Explicit one-to-one correspondence between biquandle colorings and quandle colorings.
result Biquandle homotopy invariants and quandle homotopy invariants are equivalent.
We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant V2, are necessary and sufficient to express any string link Vassiliev invariant of order two. Explicit combinatorial formulas are given for these invariants. This result is appli…
New invariant for links in 3-sphere computed and computed using diagrams.
problem Computing invariants of links in 3-sphere.
method Defining and computing the parabolic Dijkgraaf-Witten invariant.
result Computed invariants of several links and partial information using diagrams.
New singquandles help distinguish certain types of links.
problem Distinguishing different types of singular links.
method Introduced G-Family of singquandles and used them to define invariants.
result These invariants can distinguish some dichromatic singular links.
New colored link invariants using multi-quandles.
problem Developing new invariants for colored links.
method Introducing multi-quandles and topological multi-quandles.
result New colored link invariants created.
New two-parameter algebra yields stronger link invariants.
problem Developing new link invariants.
method Introducing a two-parameter bt-algebra and its associated invariant.
result The new invariant is more powerful than existing ones.
Study annular concordance invariants and refine transverse link invariants.
problem Understanding annular concordance and transverse links.
method Defined and studied annular concordance invariant, introduced modified chain complex, and refined transverse invariant.
result Refined transverse invariant and studied its relationship with transverse and braid monodromy properties.
Grid homology invariant proved for lens space links.
problem Proving combinatorial invariance of grid homology for lens space links.
method Combining combinatorial methods with sign assignments to prove invariance.
result Grid homology is a link invariant for lens space links.
New invariant counts SU(2) representations for links.
problem Counting irreducible SU(2) representations for links.
method Signed count of irreducible representations with fixed traces.
result Invariant is a sum of multivariable signatures for specific links.
Invariants from biquasile colorings distinguish surface-links.
problem Counting and distinguishing surface-links.
method Coloring oriented surface-links using biquasiles and marked graph diagrams.
result Invariants can distinguish closed surface-links and cobordisms.
Defines linking number and Milnor invariants, their properties.
problem None explicitly stated; definitions and properties are discussed.
method Definitions and properties of linking number and Milnor invariants are discussed.
result Definitions and properties of linking number and Milnor invariants are established.
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.
Study on oriented disingquandles for distinguishing singular links.
problem Distinguishing singular links of different colors.
method Introduced oriented dichromatic singular links and oriented disingquandles to define invariants.
result Found invariants that can distinguish some singular links.
Paper discusses biquandle cohomology and invariants for surface-links.
problem Computing invariants for surface-links using biquandle theory.
method Developed (co)homology theory of biquandles, introduced new invariants using broken surface diagrams and marked graph diagrams.
result Constructs new invariants for surface-links using biquandle theory.
New formulas link Milnor invariants to Heegaard Floer homology.
problem Understanding Milnor invariants through Heegaard Floer homology.
method Established new relationships between Milnor invariants and Heegaard Floer homology.
result Formula for the Milnor triple linking number from the link Floer complex.
We define a family of link concordance invariants {sn}n=2,3,⋯. These link concordance invariants give lower bounds on the slice genus of a link L. We compute the slice genus of positive links. Moreover, these invariants give lower bounds on the link splitting number of a link. Especially, t…
Study Heegaard Floer homology for two-component L-space links with zero linking number.
problem Characterize and calculate invariants for two-component L-space links with zero linking number.
method Use Heegaard Floer homology, Sato-Levine invariant, and Casson invariant to describe the relationship between these invariants and surgeries on links.
result Explicitly describe the relationship between the h-function, Sato-Levine invariant, and Casson invariant for two-component L-space links with zero linking number. New quantum invariants for 3-manifolds with boundary defined via virtual links.
problem Quantum invariants for 3-manifolds with non-vacuous boundaries.
method Surgery on manifolds of the form FimesI and virtual link diagrams. result First meaningful, nontrivial, calculable quantum invariants of 3-manifolds with non-vacuous boundary.
Paper provides a condition to distinguish links up to 4-moves.
problem Distinguishing links up to 4-moves is generally difficult.
method Defined a quotient of the fundamental group invariant preserved by 4-moves.
result Provides a necessary condition for triviality up to 4-moves.
Extends Milnor invariants to surface-links using cut-diagrams.
problem Tackles invariants for surface-links in 4-space.
method Introduces cut-diagrams and groups associated to them.
result Yields concordance and link-homotopy invariants for surface-links.
Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
Functoriality proved for colored link invariants.
problem Link and tangle invariants functoriality proof.
method Functoriality proved for colored Khovanov-Rozansky invariants.
result Functoriality of colored link homologies proved.
Characters from logarithmic VOAs linked to torus link invariants.
problem Understanding characters of logarithmic vertex operator algebras.
method Relating characters to coloured Jones invariants of torus links.
result Characters of logarithmic VOAs are limits of coloured Jones invariants of torus links.
We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite t…
Defines an odd analog of Plamenevskaya's invariant for transverse links.
problem No specific problem stated; focuses on extending an invariant.
method Defines and analyzes an odd analog invariant in Khovanov homology.
result The odd analog invariant is an invariant of transverse links with similar properties.
Study of concordances between links in 4-space, defining new invariants.
problem Investigate the set of all embedded concordances between two fixed links in 4-space.
method Define Milnor-type invariants of the set of concordances, modulo indeterminacy.
result For slice links, these invariants classify the set of concordances up to link-homotopy.
Abstract summarizes level-rank duality in knot and link invariants.
problem Distinguishing torus knots and links from hyperbolic ones.
method Chern-Simons theory and tables of knot invariants.
result Criterion to distinguish torus knots and links from hyperbolic ones.
Virtual index cocycles reformulate virtual link invariants.
problem No specific problem stated; focuses on reformulation.
method Using virtual index cocycles to reformulate invariants.
result Unified reformulation of virtual link invariants.