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 type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.
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 surgeries, which provides a filtration of link homotopy classes.
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 or less vanish, then Milnor numbers of length 2k+1 or less are well-defined. Moreover we prove that two clover links whose Milnor numbers of length k or less vanish are equivalent up to edge-homotopy and $C_{2k+1}$-equivalence if and only if those Milnor numbers of length 2k+1 or…
Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the classical case. Unlike previous attempts at generalizing link homotopy to spatial graphs, our new relation allows analogues of some standard link homotopy results and invariants. In particular we can define a type of Milnor group for a spatial graph under component homotopy, and this group determines whether or not the spatial graph is splittable. More surpri…
A handlebody-link is a disjoint union of embeddings of handlebodies in $S^3$ and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this paper, we construct a family of invariants for HL-homotopy classes of general handlebody-links, by using Milnor's link-homotopy invariants. Moreover, we give a bijection between the set of HL-homotopy classes of almost trivial handlebody-links an…
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 Alexander and Jones invariants are known for quite a while, a similar treatment of Milnor's mu-bar-invariants was missing. We fill this gap by deducing simple skein relations for link-homotopy mu-invariants of string links.
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 $\barμ$. Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanophrases. We also generalize $\barμ$ to the set of those nanophrases that correspond to virtual links.
In a previous paper, the authors proved that Milnor link-homotopy invariants modulo $n$ classify classical string links up to $2n$-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 $2n$-move and self-crossing virtualization, and up to $V^{n}$-move and self-crossing virtualization, respectively.
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain component of a link with fixing the rest components, in other words, homotopy and Delta-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition…
Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from a link $L$ to itself forms a group. In order to investigate these groups, we define Milnor-type invariants of $\mathcal{C}(L)$, which are integers defined modulo a certain indeterminacy given by Milnor invariants of $L$. We show in particular that, for a slice link $L$, these invariants classify $\mathcal{C}(L)$ up to link-homotopy.
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. Lin showed that two string links are a link-homotopc if and only if their Milnor invariants with $r=1$ coincide. This gives us that a link in $S^3$ is link-homotopic to a trivial link if and only if the all Milnor invariants of the link with $r=1$ vanish. Although Milnor invaria…
In this paper, we introduce two functions such that the subtraction corresponds to the Milnor's triple linking number; the addition obtains a new integer-valued link homotopy invariant of $3$-component links. We also have found a series of integer-valued invariants derived from four terms whose sum equals the Milnor's triple linking number. We apply this structure to give invariants of $3$-bouquet graphs.
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 type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas fo…
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 $\barμ$--invariants. We prove that, analogously as for $\barμ$--invariants, certain residue classes of tree invariants yield link homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor's indeterminacy. For practical purp…
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples of non-splittable spatial graphs up to edge (resp. vertex)-homotopy, all of whose constituent links are link-homotopically trivial.
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 invariant which detects the invertibility for some links.
Two string links are equivalent up to $2n$-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo $n$. Moreover, the set of the equivalence classes forms a finite group generated by elements of order $n$. The classification induces that if two string links are equivalent up to $2n$-moves for every $n>0$, then they are link-homotopic.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split links. In this paper, we construct some new edge (resp. vertex)-homotopy invariants of spatial graphs without any restriction of linking numbers of the constituent 2-component links by applying the generalized Sato-Levine invariant.
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 the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self Delta-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link L is link-homotopic to the unlink if and only if a link L with…
J.-B. Meilhan and the second author showed that any Milnor $\barμ$-invariant of length between 3 and $2k+1$ can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all $\barμ$-invariants of length $\leq k$ vanish. They also showed that their formula does not hold for length $2k+2$. In this paper, we improve their formula to give the $\barμ$-invariants of length $2k+2$ by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by $\barμ$-invariants of length $k+1$. …
A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, the first author proved that the restriction to Brunnian links of any Goussarov-Vassiliev finite type invariant of (n+1)-component links of degree<2n is trivial. The purpose of this paper is to study the first nontrivial case. We will show that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the Milnor link-homotopy invariants of length n+1.
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 those of length $k+1$ survive. For a clover link $c$ with the Milnor numbers of length at most $k$ vanishing, we show that the Milnor number $μ_c(I)$ for a sequence $I$ is well-defined up to the greatest common devisor of $μ_{c}(J)'s$, where $J$ is a subsequence of $I$ obtained b…
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 invariants, which are classical invariants generalizing the linking number, providing some new topological insight into quantum invariants. More precisely, we define a reduction of the universal sl_2 invariant, and show how it is captured by Milnor concordance invariants. We also show h…
Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's mu and bar mu invariants to welded and virtual links. We conclude this paper with several examples, and compute the mu invariants using the Magnus expansion and Polyak's skein relation for the mu invariants.
We extend Milnor's mu-invariants of link homotopy to ordered (classical or virtual) tangles. Simple combinatorial formulas for mu-invariants are given in terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves corresponds to axioms of Loday's diassociative algebra. The relation of tangles to diassociative algebras is formulated in terms of a morphism of corresponding operads.
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 combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the `first non vanishing' Milnor invariants can be always represented as such a linear combination.
We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the triviality of these weight systems is the presence of a certain kind of spanning tree in the intersection graph of a chord diagram.
K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space $K_ω.$ The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.
In his 1957 paper, John Milnor introduced link invariants which measure the homotopy class of the longitudes of a link relative to the lower central series of the link group. Consequently, these invariants determine the lower central series quotients of the link group. This work has driven decades of research with profound influence. One of Milnor's original problems remained unsolved: to extract similar invariants from the transfinite lower central series of the link group. We reformulate and extend Milnor's invariants in the broader setting of 3-manifolds, with his original invariants as spe…
We prove a geometric refinement of Alexander duality for certain 2-complexes, the so-called gropes, embedded into 4-space. This refinement can be roughly formulated as saying that 4-dimensional Alexander duality preserves the disjoint Dwyer filtration. In addition, we give new proofs and extended versions of two lemmas of Freedman and Lin which are of central importance in the A-B-slice problem, the main open problem in the classification theory of topological 4-manifolds. Our methods are group theoretical, rather than using Massey products and Milnor μ-invariants as in the original proofs.
We consider knotted annuli in 4-space, called 2-string-links, which are knotted surfaces in codimension two that are naturally related, via closure operations, to both 2-links and 2-torus links. We classify 2-string-links up to link-homotopy by means of a 4-dimensional version of Milnor invariants. The key to our proof is that any 2-string link is link-homotopic to a ribbon one; this allows to use the homotopy classification obtained in the ribbon case by P. Bellingeri and the authors. Along the way, we give a Roseman-type result for immersed surfaces in 4-space. We also discuss the case of ri…
We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4-manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating pi_1-null surfaces in 4-manifolds. In our applications to link-homotopy, grope splitting serves as a geometric substitute for the Milnor group.
A derivative of an algebraically slice knot $K$ is an oriented link disjointly embedded in a Seifert surface of $K$ such that its homology class forms a basis for a metabolizer $H$ of $K$. We show that for a genus three algebraically slice knot $K$, the set $\{ \barμ_{\{γ_1,γ_2,γ_3\}}(123) - \barμ_{\{γ'_1,γ'_2,γ'_3\}}(123)| \{γ_1,γ_2,γ_3\}$ and $\{γ'_1,γ'_2,γ'_3\}$ are derivatives of $K$ associated with a metabolizer $H\}$ contains $n\cdot \mathbb{Z}$ where $n$ is determined by a Seifert form of $K$ and a metabolizer $H$. As a corollary, we show that it is possible to realize any integer as a …
We initiate the homotopical study of racks and quandles, two algebraic structures that govern knot theory and related braided structures in algebra and geometry. We prove analogs of Milnor's theorem on free groups for these theories and their pointed variants, identifying the homotopy types of the free racks and free quandles on spaces of generators. These results allow us to complete the stable classification of racks and quandles by identifying the ring spectra that model their stable homotopy theories. As an application, we show that the stable homotopy of a knot quandle is, in general, mor…
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 characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link in 3-space is the degree of its associated Gauss map from the 2-torus to the 2-sphere. When the pairwise linking numbers are all zero, we giv…
Texas Christian University（德克萨斯基督教大学）；University of Pennsylvania（宾夕法尼亚大学）；University of Louisiana at Lafayette（路易斯安那大学拉斐特分校）；University of Oregon（俄勒冈大学）；University of the Virgin Islands（维尔京群岛大学）
本文提出链环补空间的 bar 上同调作为同调球中链环的新不变量，它构成 Hopf 代数，可通过曲面及其交曲线计算，并定义了张量 Hopf 代数的典范子空间，编码并超越 Milnor 不变量。
We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there is a degree-one map f:M --> M throwing K onto J and mapping M-K to M-J. These invariants are invariants of concordance and of J-characteristicness when f induces the identity on the fundamental group of M, and may be viewed as extensions of Milnor's invariants. We do not require the knots considered here to be rationally null-homologous or framed.
We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres…
Milnor's $\barμ$-invariants of links in the $3$-sphere $S^3$ vanish on any link concordant to a boundary link. In particular, they are trivial on any knot in $S^3$. Here we consider knots in thickened surfaces $Σ\times [0,1]$, where $Σ$ is closed and oriented. We construct new concordance invariants by adapting the Chen-Milnor theory of links in $S^3$ to an extension of the group of a virtual knot. A key ingredient is the Bar-Natan $\textit{Zh}$ map, which allows for a geometric interpretation of the group extension. The group extension itself was originally defined by Silver-Williams. Our ext…
In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in \mathbb{Z}^+$ and $r \geq 2$. We describe the link $L_F$ as a graph manifold using its natural open book decomposition, related to the Milnor fibration of the map-germ $f \bar{g}$ and the description of its monodromy as a quasi-periodic diffeomorphism through its Nielsen invariants. Furthermore, such a germ $F$ gives rise to a Milnor fibration $\frac{F}{|F|…
Ribbon 2-knotted objects are locally flat embeddings of surfaces in 4-space which bound immersed 3-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. In this paper we consider ribbon tubes and ribbon torus-links, which are natural analogues of string links and links, respectively. We show how ribbon tubes naturally act on the reduced free group, and how this action classifies ribbon tubes up to link-homotopy, that is when allowing each component to cross itself. At the combinatorial le…
We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [1]. Dynamical thickening leads to a conceptually simple and short proof.
We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $μ$--…
Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $\mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of $\mathbb{R}$ in $\mathbb{R}^n$ with fixed behavior outside a compact set and such that the images of the copies of $\R$ are disjoint -- even for $n=3…
The purpose of the paper is twofold. First, we give a short proof using the Kontsevich integral for the fact that the restriction of an invariant of degree 2n to (n+1)-component Brunnian links can be expressed as a quadratic form on the Milnor mu-bar link-homotopy invariants of length n+1. Second, we describe the structure of the Brunnian part of the degree 2n-graded quotient of the Goussarov--Vassiliev filtration for (n+1)-component links.
This work continues the study of a homotopy-theoretic construction of the author inspired by the Bott-Taubes integrals. Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots. Their techniques were later used by Cattaneo et al. to construct real "Vassiliev-type" cohomology classes in spaces of knots in higher-dimensional Euclidean space. By doing this integration via a Pontrjagin-Thom construction, we constructed cohomology classes in the knot space with arbitrary coefficients. We later showed that a refinement of this …
This article surveys the use of configuration space integrals in the study of the topology of knot and link spaces. The main focus is the exposition of how these integrals produce finite type invariants of classical knots and links. More generally, we also explain the construction of a chain map, given by configuration space integrals, between a certain diagram complex and the deRham complex of the space of knots in dimension four or more. A generalization to spaces of links, homotopy links, and braids is also treated, as are connections to Milnor invariants, manifold calculus of functors, and…
We define a generalization of virtual links to arbitrary dimensions by extending the geometric definition due to Carter et al. We show that many homotopy type invariants for classical links extend to invariants of virtual links. We also define generalizations of virtual link diagrams and Gauss codes to represent virtual links, and use such diagrams to construct a combinatorial biquandle invariant for virtual $2$-links. In the case of $2$-links, we also explore generalizations of Fox-Milnor movies to the virtual case. In addition, we discuss definitions extending the notion of welded links to h…
The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of B…
In this article, we study the topology of the family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ given by $F(x,y,z)=\bar{xy}(x^p+y^q)+z^r$ with $p,q,r \in \mathbb{N}$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ has isolated singularity at 0 and gives rise to a Milnor fibration $\frac{F}{|F|} \colon \mathbb{S}^{5} \setminus L_F \to \mathbb{S}^{1}$. We describe the link $L_F$ as a Seifert manifold and we show that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of $\mathbb{S}^{5}$ …
In this sequel we extend the derivation of the third order helicity to magnetic fields supported on unlinked domains in 3-space. The formula is expressed in terms of generators of the deRham cohomology of the configuration space of three points in $\R^3$, which is a more practical domain from the perspective of applications. It also admits an ergodic interpretation as an average asymptotic Milnor $\barμ_{123}$-invariant and allows us to obtain the $L^2$-energy bound for the magnetic field. As an intermediate step we derive an integral formula for Milnor $\barμ_{123}$-invariant for parametrized…
