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 …
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48 results for “Milnor's classification”
New method classifies spin 4-manifolds using Kervaire-Milnor invariant.
problem Classifying spin 4-manifolds up to stabilisation.
method Using Kervaire-Milnor invariant to compute Arf invariants and classify.
result New stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
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.…
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.
In a previous paper, the authors proved that Milnor link-homotopy invariants modulo classify classical string links up to -move and link-homotopy. As analogues to the welded case, in terms of Milnor invariants, we give here two classifications of welded string links up to -move and self-crossing virtualizat…
Study open 3-manifolds as sums of closed ones, finding a classification.
problem Classifying open 3-manifolds that are sums of closed 3-manifolds.
method Introduced topological invariants, classified when finitely many summands up to diffeomorphism.
result Unified classification of open 3-manifolds and closed 3-manifolds.
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.
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 …
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
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.
This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Wh…
We give a classification of -component links up to -move. In order to prove this classification, we characterize Brunnian links, and have that a Brunnian link is ambient isotopic to a band sum of trivial link and Milnor's links.
We define a diffeology on the Milnor classifying space of a diffeological group , constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any princip…
Study homotopy types of free racks and quandles, proving analogs of Milnor's theorem.
problem Understanding the homotopy types of free racks and quandles.
method Proved analogs of Milnor's theorem for racks and quandles and their pointed variants.
result Identified the homotopy types of free racks and quandles on spaces of generators.
Simplified proof classifies surfaces using normal curves.
problem Classifying surfaces using algebraic topology invariants.
method Using normal curves and analogy to 3-manifolds classification.
result Simple proof without using algebraic topology invariants.
Several proofs of Fáry--Milnor theorem are presented.
problem Fáry--Milnor theorem
method Sketches several proofs
result Proofs of Fáry--Milnor theorem
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.
In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.
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.
Higher-dimensional Milnor frames are characterized and contrasted with 3D Heisenberg and 4D nilpotent Lie algebras.
problem Characterizing higher-dimensional Milnor frames and their properties.
method Definition and classification of higher-dimensional Milnor frames and their relationship to known Lie algebras.
result Higher-dimensional Milnor frames are isomorphic to direct sums of 3D Heisenberg and 4D nilpotent Lie algebras and an abelian Lie algebra.
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…
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.
Paper solves Dirichlet problem at infinity for Riemannian cones.
problem Solvability of Dirichlet problem at infinity in Riemannian cones.
method Separation of variables and comparison arguments for ODE's.
result Sufficient condition for solvability related to Milnor's classification.
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…
We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the repre…
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.
Study the topology of Milnor boundaries for real analytic map germs.
problem Topology of Milnor boundaries for real analytic map germs.
method Prove Milnor boundaries are double of Milnor tubes and use generalized open-book decompositions.
result Prove Euler characteristic formulae connecting Milnor boundaries and links.
We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link and of several other links. Moreover, we refine the original Milnor invariants …
Two string links are equivalent up to -moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo . Moreover, the set of the equivalence classes forms a finite group generated by elements of order . The classification induces that if two string links are equivalent u…
Extends Milnor's criterion to biharmonic functions.
problem Deciding surface type for biharmonic functions.
method Generalizes Milnor's criterion to biharmonic functions.
result Characterizes whether a surface is hyperbolic or parabolic for biharmonic functions.
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.
New combinatorial model for Milnor fibration using oriented matroids.
problem Understanding the homotopy type of Milnor fibers of complexified real arrangements.
method Introducing a poset quasi-fibration based on a subdivision of the Salvetti complex and an oriented matroid.
result Homotopy type of Milnor fiber depends only on the combinatorial structure of the oriented matroid.
Defines Milnor number for foliations and shows its topological invariance.
problem Defining and proving invariance of Milnor number for non-isolated singularities of holomorphic foliations.
method Defining Milnor number as intersection number of sections; proving invariance via topological equivalences.
result Milnor number is invariant under topological equivalences.
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.
Milnor fibrations have been studied since 1960's. In this paper, we study singular points of differentiable maps, called Milnor fibration product maps, obtained by several Milnor fibrations. We give a characterization of singular points of such product maps, and for the case of certain weighted homogeneous polynomials,…
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…
Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.
problem Computing the first homology group of the Milnor fiber boundary for generic hyperplane arrangements.
method Analyzing the Milnor fiber boundary for hyperplane arrangements in C^3.
result Affirmative answer to the conjecture of Suciu and example of arrangements with non-trivial torsion.
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.
Study volume growth in Milnor fibers using real Lagrangians.
problem Volume growth in Milnor fibers of Brieskorn polynomials.
method Investigate real Lagrangians, use Smith inequality for involutions in wrapped Floer homology.
result Uniform lower bound of volume growth for a class of Brieskorn polynomials.
New method normalizes Milnor fibrations for real analytic maps.
problem Existence of normalized Milnor fibrations for real analytic maps.
method Introducing a homeomorphism to transform a non-normalized Milnor fibration into a normalized one.
result Normalized map defines a smooth locally trivial fibration on the sphere.
New method determines arrangement combinatorics from Milnor fiber boundary.
problem Determining arrangement combinatorics from Milnor fiber boundary.
method Explicit method using plumbing graph in normal form.
result Milnor fiber boundary determines arrangement combinatorics.
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.
A Formula For the Homology Groups of the Milnor Fibres of an Isolated Real Hypersurface Singularitymath.AG
Formula calculates homology groups of Milnor fibres for real hypersurface singularities.
problem Calculating homology groups of Milnor fibres for real hypersurface singularities.
method Established a formula for homology groups of Milnor fibres relative to their boundaries.
result Formula provides a method to compute homology groups of Milnor fibres.
Implemented Habegger-Lin algorithm for 4- and 5-component links.
problem Determining link-homotopy of links.
method Explicit computation of group actions and implementation of algorithm.
result Found new pairs of links not distinguishable by Milnor's invariants.
In this paper we extend the characterization of trivial map-germs for the real Milnor fibrations started by Church and Lamotke. Our main result cover all cases on the three dimensional real Milnor fibers.
The paper proves uniform stable radius and Milnor number equality for specific mappings.
problem Proving uniform stable radius and Milnor number equality for specific mappings.
method Analytic family construction and Newton polyhedra analysis.
result Milnor numbers of non-degenerate isolated complete intersection singularities are equal.
Real Milnor fibres become contractible after attaching handles, matching classical results.
problem Topology of real isolated hypersurface singularities.
method Proving real Milnor fibres contractible after attaching handles, defining real vanishing cycles.
result Integer homology groups of real Milnor fibres are isomorphic to those of a bouquet of spheres under certain conditions.
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.