A diagrammatical characterization of Milnor invariants

Abstract

The goal of this paper is to give a diagrammatical characterization of the information given by the Milnor invariants of links and string links. More precisely, we describe when two string links have equal Milnor invariants of length ≤ q and when a link has trivial Milnor invariants of lenght ≤ q. This will be done through the use of welded knot theory, involving the notions of arrow calculus and wq-concordance introduced by J-B. Meilhan and A. Yasuhara. These results is to be compared to the classification of links up to Cq-concordance obtained by J. Conant, R. Schneiderman and P. Teichner.

k-reduced groups and Milnor invariants

“…This isomorphism suggests that welded theory provides a sensible diagrammatic counterpart of the algebraic constructions underlying Milnor invariants. Our main theorem also refines a recent result of B. Colombari [7], who gave a diagrammatic characterization of string links having same Milnor invariants of length ≤ q. We note that the geometric properties of Milnor link invariants µ(I) with r(I) ≤ k was previously investigated in [9,27,26], using clasper theory.…”

“…By Theorems 2.8 and 3.19, welded string links are classified up to self w k -equivalence by their k-reduced longitudes. Similar phenomena occur in the classifications up to self-virtualization [1] and w k -concordance [7], and these results were extended to the case of welded links in [3] and [7] respectively, in terms of (adaptations of) the peripheral system. In this final section, we outline how a similar extension can be derived for links up to self w k -concordance.…”

“…In order to conclude the proof, it is now enough to show that any welded string link is w (k+1) -concordant to a string link with a sorted presentation, since Milnor invariants µ(I) with r(I) ≤ k are invariants of w (k+1)concordance. This can be seen as a direct corollary of [7,Cor. 2.5], which tells us that any w-tree presentation can be deformed into a sorted one by a sequence of welded concordance and surgeries along w nk+1 -trees.…”

“…Remark 3.22. Using the result of Colombari [7] mentioned in the introduction, the proof of Proposition 3.21 can be adapted in a straightfoward way to show that, for all k ≥ 1, we have an isomorphism…”

“…We address the (welded) link case in the final section of this paper. As developped there, building on the proof of our main theorem for string links, and using straightforward adaptations of the above mentioned work of Colombari [7], we obtain in particular the following for classical links.…”