FINITE TYPE LINK HOMOTOPY INVARIANTS OF k-TRIVIAL LINKS

2003 ◽  
Vol 12 (03) ◽  
pp. 375-393 ◽  
Author(s):  
JAMES R. HUGHES
Keyword(s):  
Finite Type ◽  
Link Group ◽  
Type Invariant ◽  
Linking Numbers ◽  
Link Homotopy ◽  
Group Automorphisms ◽  
Homotopy Invariants

In a recent paper [8], Xiao-Song Lin gave an example of a finite type invariant of links up to link homotopy that is not simply a polynomial in the pairwise linking numbers. Here we present a reformulation of the problem of finding such polynomials using the primary geometric obstruction homomorphism, previously used to study realizability of link group automorphisms by link homotopies. Using this reformulation, we generalize Lin's results to k-trivial links (links that become homotopically trivial when any k components are deleted). Our approach also gives a method for finding torsion finite type link homotopy invariants within "linking classes," generalizing an idea explored earlier in [1] and [10], and yielding torsion invariants within linking classes that are different from Milnor's invariants in their original indeterminacy.

scholarly journals FINITE TYPE LINK HOMOTOPY INVARIANTS

1999 ◽  
Vol 08 (06) ◽  
pp. 773-787 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
Linking Numbers ◽  
String Links ◽  
Link Homotopy ◽  
Homotopy Invariants

In [2], Bar-Natan used unitrivalent diagrams to show that finite type invariants classify string links up to homotopy. In this paper, I will construct the correct spaces of chord diagrams and unitrivalent diagrams for links up to homotopy. I will use these spaces to show that, far from classifying links up to homotopy, the only rational finite type invariants of link homotopy are the linking numbers of the components.

scholarly journals FINITE TYPE INVARIANTS AND MILNOR INVARIANTS FOR BRUNNIAN LINKS

2008 ◽  
Vol 19 (06) ◽  
pp. 747-766 ◽  
Author(s):  
KAZUO HABIRO ◽  
JEAN-BAPTISTE MEILHAN
Keyword(s):  
Quadratic Form ◽  
Finite Type ◽  
Finite Type Invariants ◽  
Type Invariant ◽  
Milnor Invariants ◽  
Link Homotopy ◽  
Homotopy Invariants ◽  
Brunnian Links

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, Habiro 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 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.

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

2000 ◽  
Vol 09 (06) ◽  
pp. 735-758 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Higher Order ◽  
Finite Type Invariants ◽  
Homotopy Invariant ◽  
Type Invariant ◽  
Link Homotopy ◽  
Homotopy 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.

scholarly journals On the Existence of Finite Type Link Homotopy Invariants

2001 ◽  
Vol 10 (07) ◽  
pp. 1025-1039 ◽  
Author(s):  
BLAKE MELLOR ◽  
DYLAN THURSTON
Keyword(s):  
Finite Type ◽  
Linking Numbers ◽  
Link Homotopy ◽  
Homotopy Invariants

We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects the errors of the first author in [11, 12].

scholarly journals FINITE TYPE LINK CONCORDANCE INVARIANTS

2000 ◽  
Vol 09 (03) ◽  
pp. 367-385 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Linking Numbers ◽  
Link Concordance ◽  
Link Homotopy

This paper is a follow-up to [10], in which the author showed that the only real-valued finite type invariants of link homotopy are the linking numbers of the components. In this paper, we extend the methods used to show that the only real-valued finite type invariants of link concordance are, again, the linking numbers of the components.

scholarly journals Tree invariants and Milnor linking numbers with indeterminacy

2020 ◽  
Vol 29 (01) ◽  
pp. 2050002
Author(s):  
R. Komendarczyk ◽  
A. Michaelides
Keyword(s):  
Recursive Procedure ◽  
Residue Classes ◽  
Linking Numbers ◽  
String Links ◽  
Link Homotopy ◽  
Homotopy Invariants ◽  
Insight Into

This paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak, which are closely related to the classical Milnor linking numbers also known as [Formula: see text]-invariants. We prove that, analogously as for [Formula: see text]-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 purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.

scholarly journals A splicing formula for the LMO invariant

2020 ◽  
pp. 1-28
Author(s):  
Gwénaël Massuyeau ◽  
Delphine Moussard
Keyword(s):  
Finite Type ◽  
Rational Homology ◽  
Three Sphere ◽  
Type Invariant ◽  
Low Degrees ◽  
Surgery Formula

Abstract We prove a “splicing formula” for the LMO invariant, which is the universal finite-type invariant of rational homology three-spheres. Specifically, if a rational homology three-sphere M is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology three-spheres, our formula expresses the LMO invariant of M in terms of the Kontsevich–LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$ . The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita’s formula for the Casson–Walker invariant, and we observe that the second term of the Ohtsuki series is not additive under “standard” splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$ , hence we get a “satellite formula” for the Kontsevich–LMO invariant.

2- AND 3-VARIATIONS AND FINITE TYPE INVARIANTS OF DEGREE 2 AND 3

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
MIGIWA SAKURAI
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Type Invariant

Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

scholarly journals HOMOMORPHIC EXPANSIONS FOR KNOTTED TRIVALENT GRAPHS

2013 ◽  
Vol 22 (01) ◽  
pp. 1250137 ◽  
Author(s):  
DROR BAR-NATAN ◽  
ZSUZSANNA DANCSO
Keyword(s):  
Finite Type ◽  
Simple Proof ◽  
Knot Theory ◽  
Large Set ◽  
Correction Factors ◽  
Connected Sums ◽  
Trivalent Graphs ◽  
Type Invariant ◽  
Knotted Trivalent Graphs ◽  
Standard Operations

It had been known since old times (works of Murakami–Ohtsuki, Cheptea–Le and the second author) that there exists a universal finite type invariant ("an expansion") Z old for knotted trivalent graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operations, it is "homomorphic"). Yet perhaps the most important operation on KTGs is the "edge unzip" operation, and while the behavior of Z old under edge unzip is well understood, it is not plainly homomorphic as some "correction factors" appear. In this paper we present two equivalent ways of modifying Z old into a new expansion Z, defined on "dotted knotted trivalent graphs" (dKTGs), which is homomorphic with respect to a large set of operations. The first is to replace "edge unzips" by "tree connected sums", and the second involves somewhat restricting the circumstances under which edge unzips are allowed. As we shall explain, the newly defined class dKTG of KTGs retains all the good qualities that KTGs have — it remains firmly connected with the Drinfel'd theory of associators and it is sufficiently rich to serve as a foundation for an "algebraic knot theory". As a further application, we present a simple proof of the good behavior of the LMO invariant under the Kirby II (band-slide) move, first proven by Le, Murakami, Murakami and Ohtsuki.

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