Related papers: Lifting Milnor Invariants for 3-Component Links
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…
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…
We use the knot homology of Khovanov and Lee to construct link concordance invariants generalizing the Rasmussen $s$-invariant of knots. The relevant invariant for a link is a filtration on a vector space of dimension $2^{|L|}$. The basic…
J.-B. Meilhan and the second author showed that any Milnor $\bar{\mu}$-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…
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 \bar{\mu}_J(L) vanishes for any sequence J with length at most k, then any Milnor…
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.…
An explicit polynomial in the linking numbers $l_{ij}$ and Milnor's triple linking numbers $\mu(rst)$ 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.…
We generalize Milnor link invariants to all types of surface-links in $4$--space (possibly with boundary). This is achieved by using the notion of cut-diagram, which is a 2-dimensional generalization of Gauss diagrams, associated to…
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…
We define numerical link-homotopy invariants of link maps of any number of components, which naturally generalize the Kirk invariant. The Kirk invariant is a link-homotopy invariant of 2-component link maps given by linking numbers of loops…
One construction of the Alexander polynomial is as a quantum invariant associated with representations of restricted quantum $\mathfrak{sl}_2$ at a fourth root of unity. We generalize this construction to define a link invariant…
In his 1957 paper, John Milnor introduced a collection of invariants for links in $S^3$ detecting higher-order linking phenomena by studying lower central quotients of link groups and comparing them to those of the unlink. These invariants,…
Polyak showed that any Milnor's $\overline{\mu}$-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…
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…
We show that Tim Cochran's invariants $\beta^i(L)$ of a $2$-component link $L$ in the $3$--sphere can be computed as intersection invariants of certain 2-complexes in the $4$--ball with boundary $L$. These 2-complexes are special types of…
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…
We define an annular concordance invariant and study its properties. When specialized to braids, this invariant gives bounds on band rank. We introduce a modified chain complex to reformulate the invariant. Then, by focusing on a special…
We show that for an $n$-component, $n$-bridge link and a positive integer $m$, the following is true: If the longitudes of $L$ lie in the $(m+2)$-th term of the lower central series of the link group then all the finite type invariants of…
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…
We extend the $sl(3)$-polynomial invariant for links to tangles. Motivated by Kuperberg's construction of this invariant via planar trivalent graphs, we first define a category of $sl(3)$ webs and its sister linear category, and describe…