Month: March 2021

Characterization(s) of the Reduced Peripheral System

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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.

Arf invariants in low dimensions

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Michael Klug (UC-Berkeley)

I will briefly discuss some work in progress regarding a relationship between several different mod 2 invariants in dimensions 2, 3, and 4. In particular, I will relate the Arf invariant of a knot, the Arf invariant of a characteristic surface, the Rochlin invariant of a homology sphere, and the Kirby-Siebenmann invariant of a 4-manifold.

Pure Braids and Link Concordance

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Miriam Kuzbary (Georgia Tech)

The knot concordance group can be contextualized as organizing problems about 3- and 4-dimensional spaces and the relationships between them. Every 3-manifold is surgery on some link, not necessarily a knot, and thus it is natural to ask about such a group for links. In 1988, Le Dimet constructed the string link concordance groups and in 1998, Habegger and Lin precisely characterized these groups as quotients of the link concordance sets using a group action. Notably, the knot concordance group is abelian while, for each n, the string link concordance group on n strands is non-abelian as it contains the pure braid group on n strands as a subgroup. In this talk, I will discuss my result the quotient of each string link concordance group by its pure braid subgroup is still non-abelian.

Movie moves for singular link cobordisms in 4-dimensional space

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Carmen Caprau (California State University-Fresno)

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

Slides for talk:

CKVK Seminar-March 1, 2021