Subject: RE: question about braid groups Date: Wed, 23 Jun 2004 12:37:57 -0400 From: "Hughes, James " To: "Don Davis" In answer to David Yetter's question about pure braids: The following facts hold in the case of link-homotopy, but I'm fairly sure they hold in the classical case as well. The subgroup of the pure braid group consisting of all pure braids such that all (pairwise) linking numbers of the closure of the braid are zero is the second lower central subgroup of the pure braid group. In general, for the n-strand pure braid group, the Brunnian braids coincide with the (n-1)-st lower central subgroup, so when n=3 the Brunnian braids and braids with all linking numbers zero do indeed coincide. James R. Hughes Associate Professor and Chair of Mathematical Sciences Elizabethtown College Elizabethtown, PA 17022 (717) 361-1334 -----Original Message----- From: Don Davis [mailto:dmd1@Lehigh.EDU] Sent: Wednesday, June 23, 2004 11:44 AM To: Don Davis Subject: question about braid groups Subject: A braid group question for the topology list Date: Mon, 21 Jun 2004 12:18:48 -0500 From: David Yetter Has anyone encountered the following subgroups of Artin's braid groups before: The subgroup of all pure braids such that the all linking numbers of the link obtained by closing the braid are all zero? These are not the Brunnian braids: for n = 2 all pure braids are Brunnian, for n = 3 the two subgroups coincide, for n > 3 the subgroup I'm looking for is larger, as it contains all tensor products of Brunnian braids (for instance at n = 6 the tensor product of two copies of a 3-braid B which represents the Borromean rings, or for that matter at n = 4 it contains 1\otimes B, B\otimes 1, and their product, none of which is Brunnian. Best regards to all, David Yetter Assoc. Prof. of Mathematics Kansas State University