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

Abstract

To a link (obtained by closing a string link) we associate a certain diagram of groups. If two links are concordant, we show that there exists a certain type of isomorphism between the group diagrams.

String links with the same closure and group diagrams

“…In this paper we consider first the case of ambient isotopy (see Theorem 4). Despite the fact that our assumption (ambient isotopy) is stronger than that in our paper [2] (concordance), we provide a different, more direct proof. The proof is based in a result from Habegger-Lin [5] (see Theorem 3 below) different from that used in [2].…”

“…Despite the fact that our assumption (ambient isotopy) is stronger than that in our paper [2] (concordance), we provide a different, more direct proof. The proof is based in a result from Habegger-Lin [5] (see Theorem 3 below) different from that used in [2]. We also provide an alternative proof for the main Theorem in [1] (see Theorem 12 below), which allows some clear geometric interpretation of that result.…”

“…As we saw in [2], since f n conjugates the classes of the meridians α i F(k) n by the classes of the correspondent longitudes (see [5]), and remembering that…”

“…Let f × 1 denote the 2k-string link obtained from a k-string link f by adding k straight strings at its end (see [2]).…”

“…We called such type of isomorphism a braid-special isomorphism (there was a small mistake in the diagram which we corrected in [2, section 4]). In [2] we used our techniques to study the case of link concordance. In this case we have n-level group diagrams, n 2, and, not like for link-homotopy, we get only that if the string links have the same closure, up to concordance, then there is, for every n 2, a n-level braid-special isomorphism between the string links n-level group diagrams.…”