Boundary String Links

Abstract

The group BSL(k) of boundary cobordism classes of boundary k-string links is defined. An epimorphism from BSL(k) to a group of cobordism classes of matrices is defined. An action of a certain group of pure braids on BSL(k) provides all possible splittings for a given boundary k-link. A necessary and sufficient condition is given for two elements of BSL(k) to have the same closure as an F(k)-link (i.e., a boundary k-link with one of its splittings), up to F(k)-cobordism.

A necessary condition for two string links to have the same closure up to concordance

“…We will denote by x i ¼ x i ð f Þ A pð f Þ, for all i A k, the top meridians of f and by y i ¼ y i ð f Þ A pð f Þ, for all i A k, the bottom meridians of f (see Fig. 3 and [2]).…”

String links with the same closure and group diagrams

“…Then A(RF(k)) is generated by {α ij }, for 1 i = j k (see [7]). It follows from [3,Proposition 6] that our map is onto.…”