Milnor's Link Research Articles

Asymptotic Lines on Real Milnor Links of a Family that Realizes Any Genus

Asymptotic Lines on Real Milnor Links of a Family that Realizes Any Genus

Site-specific Gordian distances of spatial graphs

A site-specific Gordian distance between two spatial embeddings of an abstract graph is the minimal number of crossing changes from one to another where each crossing change is performed between two previously specified abstract edges of the graph. It is infinite in some cases. We determine the site-specific Gordian distance between two spatial embeddings of an abstract graph in certain cases. It has an application to puzzle ring problem. The site-specific Gordian distances between Milnor links and trivial links are determined. We use covering space theory for the proofs.

Invariants and structures of the homology cobordism group of homology cylinders

The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants and Johnson homomorphisms. They give rise to a descending filtration of the homology cobordism group of homology cylinders. We show that each successive quotient of the filtration is free abelian of finite rank. Second, we define Hirzebruch-type intersection form defect invariants obtained from iterated p-covers for homology cylinders. Using them, we show that the abelianization of the intersection of our filtration is of infinite rank. Also we investigate further structures in the homology cobordism group of homology cylinders which previously known invariants do not detect.

Whitney tower concordance of classical links

This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new quadratic refinement of the intersection theory is introduced, measuring Whitney disk framing obstructions. It is shown that the filtrations are completely classified by Milnor invariants together with new higher-order Sato-Levine and higher-order Arf invariants, which are obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link in the 3-sphere. Applications include computation of the grope filtration, and new geometric characterizations of Milnor's link invariants.

HAMILTONIAN BF THEORY AND PROJECTED BORROMEAN RINGS

It is shown that the canonical formulation of the Abelian BF theory in D = 3 allows to obtain topological invariants associated to curves and points in the plane. The method consists on finding the Hamiltonian on-shell of the theory coupled to external sources with support on curves and points in the spatial plane. We explicitly calculate a nontrivial invariant that could be seen as a "projection" of the Milnor's link invariant [Formula: see text], and as such, it measures the entanglement of generalized (or projected) Borromeans Rings in the Euclidean plane.

THE TOPOLOGICAL THEORY OF THE MILNOR INVARIANT $\bar\mu(1,2,3)$

We study a topological Abelian gauge theory that generalizes the Abelian Chern–Simons one, and that leads in a natural way to the Milnor's link invariant [Formula: see text] when the classical action on-shell is calculated.

Classification of n-component Brunnian links up to [formula omitted]-move

We give a classification of n-component links up to C n -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 a trivial link and Milnor's links.

Rigidity of CR-immersions into Spheres

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion $f\colon (M,p)\to \mathbb S$ and $d < n/2$, then $f$ is {\em rigid} in the sense that any other immersion of $(M,p)$ into $\mathbb S$ is of the form $\phi\circ f$, where $\phi$ is a biholomorphic automorphism of the unit ball $\mathbb B\subset\bC^{n+d+1}$. As an application of this result, we show that an isolated singularity of an irreducible analytic variety of codimension $d$ in $\bC^{n+d+1}$ is uniquely determined up to affine linear transformations by the local CR geometry at a point of its Milnor link.

Links, pictures and the homology of nilpotent groups

We give a geometric slice-like characterization for the vanishing of Milnor's link invariants by proving the k-slice conjecture. This conjecture states that a link L has vanishing Milnor μ ̄ -invariants of length ⩽2 k if and only if L bounds disjoint surfaces in a four disk in such a way that the fundamental group of the complement admits free nilpotent quotients of class k. In the course of our proof, we compute the dimension ⩽3 homology groups of finitely generated free nilpotent Lie rings and groups. We develop a new algorithm for constructing a weighted chain resolution for a nilpotent group with torsion free lower central series quotients, and with the property that its associated graded complex is the Koszul complex of the associated graded Lie ring. This give a new derivation of the May spectral sequence relating the group homology of the nilpotent group to the Lie ring homology of its associated graded Lie ring. Finally, we define μ ̄ -invariants of “pictures” and use these to describe a generating set of cocycles in the cohomology of the free nilpotent groups. Some sample computations follow.

FINITE TYPE LINK HOMOTOPY INVARIANTS II: Milnor's $\bar\mu$-Invariants

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's link homotopy invariant [Formula: see text] is a finite type invariant, of type 1, in this sense. We also generalize this approach to Milnor's higher order [Formula: see text] invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

Milnor's link invariants attached to certain Galois groups over $\mathbf {Q}$

This is a résumé of the author's recent work on certain analogies between primes and links. The purpose of this article is to introduce a new invariant, called Milnor invariant, in algebraic number theory, based on an analogy between the structure of a certain Galois group over the rational number field and that of the group of a link in three dimensional Euclidean space. It then turns out that the Legendre, Rédei symbols are interpreted as our link invariants. We expect that this is a tip of an arithmetical theory after the model of link theory which may give a new insight in algebraic number theory. The details will be published elsewhere.