arXiv research

Defines linking number and Milnor invariants, their properties.

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 …

Study Milnor invariants for covering links to distinguish certain links.

Paper introduces simplified formulas for Milnor's triple linking number.

New formulas link Milnor invariants to Heegaard Floer homology.

New invariant refines Milnor's triple linking number, revealing more information for complex links.

Paper introduces new link homotopy invariants and applies them to 3-bouquet graphs.

Formula for Milnor triple linking number in link diagrams with multiple crossings.

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 $2k+1$ are well-defined if those of length at most $k$ vanish, and that the Milnor numbers of length at least $2k+2$ are not well-defined if …

Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.

The paper proves link homotopy invariants from tree invariants and explains indeterminacy.

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 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…

We study relations between the Alexander-Conway polynomial $\nabla_L$ and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of $\nabla_L$ of an m-component link L all of whose Milnor numbers $μ_{i_1... i_p}$ van…

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.…

Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.

Defines new link-homotopy invariants using Milnor's higher order link invariants.

Study of Milnor invariants and ropelength of spherical links.

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…

New invariants for handlebody-links using Milnor's invariants.

New Milnor's invariant condition for topologically slice links.

The paper studies triple linking numbers of genus three knots and their derivatives.

New obstructions show links with vanishing Milnor invariants may not be concordant to homology boundary links.

The paper provides estimates for embedding thickness of links using finite type invariants.

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 $μ$ invariant. These invariants are seen to be naturally r…

Polyak showed that any Milnor's $\overlineμ$-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 $μ$-invariants. We sho…

Generalizes Milnor's invariants to knots in 3-manifolds.

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…

Combines combinatorial method to extend Milnor invariants to welded links.

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…

Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previo…

To each three-component link in the 3-sphere, we associate a geometrically natural characteristic map from the 3-torus to the 2-sphere, and show that the pairwise linking numbers and Milnor triple linking number that classify the link up to link homotopy correspond to the Pontryagin invariants that classify its charact…

Extends Milnor's invariants to knots and links in 3-manifolds.

Reconfigures Milnor invariants using unipotent Magnus embeddings.

Explains Milnor invariants for links and surfaces.

Homotopy equivalence found between Milnor-Lê fibers of specific singularities.

Study Stein and Milnor fillings of links from surface singularities.

Study on deformations of polynomials and their Milnor fibration topology.

Study shows boundary of Milnor fibre is invariant for certain singularities.

An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $μ(rst)$ on six component links is shown to be a well-defined finite type link-homotopy invariant. This solves a problem raised by B. Mellor and D. Thurston. An extension of our construction also produces a finite type link invar…

Study new invariants for string links using Milnor and Morita-Milnor homomorphisms.

We show that the Casson knot invariant, linking number and Milnor's triple linking number, together with a certain 2-string link invariant $V_2$, 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…

Extends Milnor's invariants to 3-manifolds, solving an open problem.

From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field of kernels of the derivative of the map. Homotopically, such a plane field deter…

This paper characterizes Milnor invariants using diagrammatic methods.

Classifies welded string links up to specific moves.

Extends Milnor invariants to surface-links using cut-diagrams.

In the present article we determine and characterize completely the support genus, the binding number and the norm of a page of an open book under the following restrictions: M is a rational homology sphere which can be realized as the link of a surface singularity. Moreover, we restrict ourselves to the collection of …