Classification of virtual string links up to cobordism

Abstract

Cobordism of virtual string links on n strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links on n strands up to cobordisms and elements of the direct product of n(n − 1) copies of the integers. This paper also shows that virtual string links up to unwelded equivalence are classified by those groups. Finally, the related theory of welded string link cobordism is defined herein and shown to be trivial for string links with one component.

k-reduced groups and Milnor invariants

“…This fact was already known, as any welded string link is invertible up to concordance[10, Prop. 6].…”

“…For k = 1, this notion turns out to coincide with the usual link-homotopy relation for string links [1]. On the other hand, there is a combinatorial equivalence relation of welded concordance for welded (string) links, which is generated by birth, death and saddle moves [5,10] and which naturally encompasses the topological concordance for classical (string) links. The self w k -concordance is the equivalence relation obtained by combining the above two relations, and our main result states that this characterizes combinatorially the information contained in Milnor invariants µ(I) with r(I) ≤ k.…”

“…The notion of concordance for welded links was introduced and studied in [5,10]. Two n-component welded string links L and L are welded concordant if one can be obtained from the other by a sequence of welded equivalence and the birth/death and saddle moves of Figure 6, such that, for each i ∈ {1, • • • , n}, the number of birth/death moves used to deform the ith component of L into the ith component of L is equal to the number of saddle moves.…”

Unrestricted virtual braids and crystallographic braid groups

“…sδ −1 k s −1 = 1 for all 2 ≤ k ≤ n(s), or in other words, δ k and s commute. It follows from (15) that γ 1 = δ 1 . sδ −1 1 s −1 , and thus…”