The Kontsevich integral and Milnor’s invariants

By N. Habegger, Gregor Masbaum. 2000

140 citations recorded by the source corpus.

Abstract A formula for computing the Milnor (concordance) invariants from the Kontsevich integral is obtained. The reduced Kontsevich integral (with values in the quotient by all loop diagrams) is shown to be the universal concordance invariant of finite type. Some applications are discussed.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/896723add0c4773aa432d90e9f06689e28b4e016

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Problems on invariants of knots and 3-manifolds (2004, background): “Habegger and Masbaum [164] showed that on the chord diagram level, the Milnor invariants (including products of Milnor invariants) are exactly the ones that vanish on Jacobi diagrams that contain internal loops, and also that the Milnor invariants are the only rational-valued finite-type invariants ”
Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant (2008, methodology): “11 given in [6] is indirect: Based on Habegger’s correspondence between Johnson homomorphisms and Milnor’s μ invariants [13], it uses the connection between the latter invariants and the Kontsevich integral [14].”
Claspers and finite type invariants of links. (2000, methodology): “results of uni-trivalent graphs are given by K Y Ng [41] and by N Habegger and G Masbaum [19].”
Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, background): “Thus, the first named author, DBN, was absolutely delighted when in January 2008 Anton Alekseev described to him his joint work [2] with Charles Torossian: Anton told DBN that they found a relationship between the Kashiwara–Vergne conjecture [52], a cousin of the Duflo isomorphism (which DBN already”
Drinfeld associators, Braid groups and explicit solutions of the Kashiwara–Vergne equations (2009, unknown): “In this section, we first show that for any O ∈ Parn+1, the isomorphism μ̃O : PBn+1(k)→ exp(t̂n+1) (see (8)) restricts to an isomorphism Fn(k)→ exp(f̂n ) (in the case of the left parenthesization, this was proved in [HM]).”
Grope cobordism of classical knots (2000, background): “the loop degree and its relation to gropes. Even though our (grope) proof is new, the following result seems well known to experts. Compare in particular the rational analogue of Habegger and Masbaum [14] and the homological version of Levine [19]. Theorem 5. If a link LΓ is obtained from a link L b”
A rational noncommutative invariant of boundary links (2001, background): “The question arises: are there any links whose Kontsevich integral has vanishing tree-part? Using Habegger-Masbaum (see [30]), this condition is equivalent to the vanishing of all Milnor invariants of a link.”