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.
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…
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.
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.
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.
In this paper, we formulate a construction of ideal coset invariants for surface-links in 4-space using invariants for knots and links in 3-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an invariant for surface-links called the Kauffman bracket ideal coset invariant of sur…
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.
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.
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.
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.
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 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.
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 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 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 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.
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 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.
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…
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.
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.
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.
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…
Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi--direct group product. By usi…
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.
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.
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.
Forbidden moves categorify fused links into quivers.
problem Categorify fused links to invariant of links.
method Use forbidden moves to categorify fused links.
result Obtain three polynomial invariants of links.
Paper introduces new link homotopy invariants and applies them to 3-bouquet graphs.
problem Milnor's triple linking number and its applications in link homotopy.
method Developed new integer-valued link homotopy invariants and applied them to 3-bouquet graphs.
result Found new integer-valued invariants derived from four terms summing to Milnor's triple linking number.
New invariant for singular links via bt-algebra.
problem Invariants for singular links.
method Representations of singular braid monoid into two parameter bt-algebra.
result More powerful than previous invariants.
Aicardi's invariant F(L) is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends F(L) to colored singular links, showing it's stronger than HOMFLY-PT polynomial. New quantum invariants for 3-manifolds with boundary calculated.
problem Quantum invariants of compact 3-manifolds with non-vacuous boundary.
method Virtual link diagrams and surgery on manifolds of the form FimesI. result Explicit calculations of invariants for some examples.
In the present paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to …
Finite type invariants separate PL links in 3D space.
problem If two PL links are isotopic, are they PL isotopic?
method Proved PL isotopic links are indistinguishable by finite type invariants.
result Finite type invariants separate PL links in 3D space if and only if certain conjectures hold.
In this paper, we give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closure of (cabled) string links. We show that finite type invariants classify string links up to …
The paper extends BPS invariants for framed knots and links.
problem Investigating BPS invariants for framed knots and links.
method Using the dual A-polynomial and framing change formula, the paper extends the relationship between algebraic curves and BPS invariants to framed knots and links.
result Explicit formulas for extremal A-polynomials and BPS invariants of framed knots, and numerical calculations for framed Whitehead links and Borromean rings.
Invariants for 3-manifolds with embedded links using Hopf algebras.
problem Constructing invariants for 3-manifolds with embedded framed links.
method Using involutory Hopf algebras and representations of Drinfeld double.
result The invariant recovers known invariants for specific cases.
We introduce the notion of a quandle with a good involution and its homology groups. Carter et al. defined quandle cocycle invariants for oriented links and oriented surface-links. By use of good involutions, quandle cocyle invariants can be defined for links and surface-links which are not necessarily oriented or orie…