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.
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48 results for “Milnor links”
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…
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.
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.
Given a clover link, we construct a bottom tangle by using a disk/band surface of the clover link. Since the Milnor number is already defined for a bottom tangle, we define the Milnor number for the clover link to be the Milnor number for the bottom tangle and show that for a clover link, if Milnor numbers of length k …
Reconfigures Milnor invariants using unipotent Magnus embeddings.
problem Computing and understanding Milnor invariants of links.
method Central group extensions and unipotent Magnus embeddings.
result First non-vanishing invariants and refinements of higher degree invariants computed.
Explains Milnor invariants for links and surfaces.
problem Milnor invariants of links and surfaces.
method Expository viewpoint of research.
result Translation and explanation of Milnor invariants.
For an -component link , the Milnor's isotopy invariant is defined for each multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here is called the length. Let denote the maximam number of times that any index appears. It is known that Milnor invariants with are link-homotopy invariant. N. Habegger and X. S.…
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.
Study Stein and Milnor fillings of links from surface singularities.
problem Comparing Stein and Milnor fillings of links from surface singularities.
method Analyzing the topology and obstructions of Stein fillings and Milnor fillings.
result Milnor fillings have bounded topology, while Stein fillings can be more varied.
Paper introduces simplified formulas for Milnor's triple linking number.
problem Computational difficulty in calculating Jones polynomial for topological polymers.
method Developed Gauss diagram formulas for Milnor's Vassiliev invariants.
result Introduced non-torsion valued Milnor's triple linking number.
Extends Milnor's invariants to 3-manifolds, solving an open problem.
problem Extract transfinite Milnor invariants for 3-manifolds.
method Develops a theory of transfinite invariants for 3-manifold groups.
result Realizes nontrivial values of transfinite Milnor invariants.
This paper characterizes Milnor invariants using diagrammatic methods.
problem Classifying links and string links based on their Milnor invariants.
method Diagrammatical characterization using welded knot theory and arrow calculus.
result Conditions for equality of Milnor invariants and triviality of certain lengths.
Classifies welded string links up to specific moves.
problem Classifying welded string links up to moves.
method Using Milnor invariants, classify welded string links up to -move and -move.
result Classifies welded string links up to -move and -move.
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.
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.
Formula for Milnor triple linking number in link diagrams with multiple crossings.
problem Calculating Milnor triple linking number for complex link diagrams.
method Polyak-Viro type formula with explicit computation of configuration space integral.
result Formula applicable to diagrams with triple or more crossings.
The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…
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 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 -component link, providing more information than classical triple linking numbers.
We describe Milnor open books and Legendrian surgery diagrams for canonical contact structures of links of some rational surface singularities. We also describe an infinite family of Milnor fillable contact 3-manifolds so that the Milnor genus (resp. Milnor norm) is strictly greater than the support genus (resp. suppor…
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.
J.P. Levine introduced a clover link to investigate the indeterminacy of the Milnor invariants of a link. It is shown that for a clover link, the Milnor numbers of length at most are well-defined if those of length at most vanish, and that the Milnor numbers of length at least are not well-defined if …
We give an example of a 3-component smoothly slice boundary link, each of whose components has a genus one Seifert surface, such that any metaboliser of the boundary link Seifert form is represented by 3 curves on the Seifert surfaces that form a link with nonvanishing Milnor triple linking number. We also give a gener…
Study of Milnor invariants and ropelength of spherical links.
problem Understanding the relationship between the thickness of spherical links and their Milnor invariants.
method Generalized Massey products and Milnor invariants to spherical links, finding optimal asymptotic bounds.
result Optimal asymptotic bounds on Milnor invariants in terms of thickness, revealing a polynomial vs exponential regime.
We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length < k are all zero into a link with vanishing Milnor invariants of length < 2k, and we provide formulas for th…
It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor's invariants with repeated indices are invariants not only of isotopy, but also of self C_k-moves. A self C_k-move is a natural generalization of link homotopy based on certain degree k clasper su…
New Milnor's invariant condition for topologically slice links.
problem Conditions for topologically slice links involving Milnor's invariant.
method Different Milnor's invariant condition to handle cases not covered by previous theorem.
result New sufficient conditions for topologically slice 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…
Characterizes Milnor invariants with limited repetitions.
problem Understanding Milnor invariants with constraints on repetition.
method Algebraic and diagrammatic approaches using -reduced free groups and welded knot theory.
result Algebraic and diagrammatic characterizations of Milnor invariants.
Open book decompositions of links of quotient surface singularities and support genus problemmath.GT
In this paper we write explicitly the open book decompositions of links of quotient surface singularities supporting the corresponding unique Milnor fillable contact structure. The page-genus of these Milnor open books are minimal among all Milnor open books supporting the same contact structure. We also investigate wh…
Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.
problem Tackles the representation of Milnor's triple linking number.
method Establishes an analogous description for Milnor's triple linking number using counts of chord diagrams and doodle invariants.
result Shows that Milnor's triple linking number can be represented in terms of chord diagrams and doodle invariants.
Proves non-solvability of concordance groups using Milnor invariants.
problem Non-solvability of concordance groups of 2-string links and strongly invertible knots.
method Using Milnor invariants to prove non-solvability.
result Proves non-solvability of and equivariant concordance groups of strongly invertible knots.
We establish several new results about both the (n)-solvable filtration, F_n^m, of the set of link concordance classes and the (n)-solvable filtration of the string link concordance group. We first establish a relationship between Milnor's invariants and links, L, with certain restrictions on the 4-manifold bounded by …
Minor typographical errors fixed. Cochran constructed many links with Alexander module that of the unlink and some nonvanishing Milnor invariants, using as input commutators in a free group and as an invariant the longitudes of the links. We present a different and conjecturally complete construction, that uses element…
New invariants lift Milnor invariants for 3-component links.
problem Classifying 3-component links using Milnor invariants.
method Defined and proved invariants , introduced , and showed their equivalence.
result Invariants lift certain Milnor invariants and are invariant under weak cobordism.
Link concordance and Whitney towers linked to Milnor invariants.
problem Link concordance and Whitney towers classification.
method Clasper surgeries, Whitney towers, and Milnor invariants.
result Link concordance and Whitney towers classified in terms of Milnor invariants.
Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.
We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant \barμ_J(L) vanishes for any sequence J with length at most k, then any Milnor \barμ-invariant \barμ_I(L) with length between 3 and 2k+1 can be represented as a c…
Polyak showed that any Milnor's -invariant of length 3 can be represented as a combination of Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called -invariants. We sho…
We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's invariant. These invariants are seen to be naturally r…
The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known as --invariants. We prove that, analogously as for --invariants, certain residue classes of tree invariants yield link homotopy invariants of c…
Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self crossing changes. Milnor introduced invariants under link homotopy called . Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanop…
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.
Floer homology linked to Milnor fibers for certain singularities.
problem Understanding Floer homology of singularities using Milnor fibers.
method Combining Floer theory with Milnor fibers, using combinatorial formulas.
result Equality of Casson invariants in Donaldson and Seiberg-Witten theories.
The Milnor degree of a 3-manifold is an invariant that records the maximum simplicity, in terms of higher order linking, of any link in the 3-sphere that can be surgered to give the manifold. This invariant is investigated in the context of torsion linking forms, nilpotent quotients of the fundamental group, Massey pro…
The universal sl_2 invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the h-adic completed tensor powers of the quantized enveloping algebra of sl_2. In this paper, we exhibit explicit relationships between the universal sl_2 invariant and Milnor invaria…
We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite ty…