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.

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 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 TREE DIAGRAMS FOR STRING LINKS II: DETERMINING CHORD DIAGRAMS

2008 ◽  
Vol 17 (06) ◽  
pp. 649-664
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Intersection Graph ◽  
Chord Diagram ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
String Links

In previous work [7], we defined the intersection graph of a chord diagram associated with a string link (as in the theory of finite type invariants). In this paper, we look at the case when this graph is a tree, and we show that in many cases these trees determine the chord diagram (modulo the usual 1-term and 4-term relations).

scholarly journals INTERSECTION GRAPHS FOR STRING LINKS

2006 ◽  
Vol 15 (01) ◽  
pp. 53-72 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Adjacency Matrix ◽  
Intersection Graphs ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
String Links ◽  
Homfly Polynomials

We extend the notion of intersection graphs for chord diagrams in the theory of finite type knot invariants to chord diagrams for string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graphs, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

BRAIDED CHORD DIAGRAMS

1998 ◽  
Vol 07 (01) ◽  
pp. 1-22 ◽  
Author(s):  
Joan S. Birman ◽  
Rolland Trapp
Keyword(s):  
Finite Type ◽  
Equivalence Relation ◽  
Chord Diagram ◽  
Finite Type Invariants ◽  
Chord Diagrams ◽  
Braid Index

The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spanning sets for the algebra of chord diagrams of braid index 3.

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