Milnor’s concordance invariants

Classification of welded links and welded string links up to w_q-concordance

June 7, 2021May 30, 2021 | chrisman.76

Boris Colombari (Aix-Marseille Université)

The notion of $w_q$ -concordance has been introduced by J-B. Meilhan and A. Yasuhara through the use of arrow calculus. It is a welded analogue of the $C_k$ -concordance on classical links coming from the clasper calculus introduced by K. Habiro. In this talk I will present a classification of welded string links (resp. welded links) up to $w_q$ -concordance by their Milnor invariants (resp. by their $q$ -nilpotent peripheral system). I will compare these results to the classification of classical links up to $C_k$ -concordance obtained by J. Conant, R. Schneiderman and P. Teichner before introducing the relevant invariants on welded objects. I will give elements of the proof of my results using a version of arrow calculus adapted to the representation of welded objects by Gauss diagrams.

Classification of string links up to 2n-moves and link-homotopy

May 31, 2021May 30, 2021 | chrisman.76

Kodai Wada (Kobe University)

$2n$

Characterization(s) of the Reduced Peripheral System

March 22, 2021March 22, 2021 | chrisman.76

Jean-Baptiste Meilhan (Université Grenoble Alpes)

The reduced peripheral system was introduced by Milnor in the 50’s for the study of links up to link-homotopy, i.e. up to homotopies leaving distinct components disjoint. This invariant, however, fails to classify links up to link-homotopy for links of 4 or more components. The purpose of this paper is to show that the topological information which is detected by Milnor’s reduced peripheral system is actually 4-dimensional. We give a topological characterization in terms of ribbon solid tori in 4-space up to link-homotopy, using a version of Artin’s Spun construction. The proof relies heavily on an intermediate characterization, in terms of welded links up to self-virtualization, providing hence a purely topological application of the combinatorial theory of welded links.

Triple linking and Heegaard Floer homology

August 24, 2020August 16, 2020 | chrisman.76

Allison Moore (Virginia Commonwealth University)

We will describe several appearances of Milnor’s link invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.

Extending Milnor’s concordance invariants to virtual knots and welded links

May 18, 2020May 18, 2020 | chrisman.76

Micah Chrisman (Ohio State University)

Milnor’s $\bar{\mu}$ -invariants for links in the 3-sphere vanish on any link concordant to a boundary link. In particular, they are are trivial for any classical knot. Here we define an analogue of Milnor’s concordance invariants for knots in thickened surfaces $\Sigma \times [0,1]$ , where $\Sigma$ is closed and oriented. These invariants vanish on any knot concordant to a homologically trivial knot in $\Sigma \times [0,1]$ . We use them to give new examples of non-slice virtual knots having trivial Rasmussen invariant, graded genus, affine index polynomial, and generalized Alexander polynomial. Moreover, we complete the slice status classification of the 2564 virtual knots having at most five classical crossings and reduce to four (of 90235) the number of virtual knots with six classical crossings having unknown slice status. Furthermore, we prove that in contrast to the classical knot concordance group, the virtual knot concordance group is not abelian. As part of the construction of the extended $\bar{\mu}$ -invariants, we also obtain a generalization of the $\bar{\mu}$ -invariants of classical links in $S^3$ to ribbon torus links in $S^4$ and welded links.