Subject: response about braid group discussion (Yetter, Hughes) Date: Fri, 23 Jul 2004 15:28:44 -0400 (EDT) From: Stefan Forcey To: dmd1@Lehigh.EDU (Don Davis) A further note/query regarding David Yetter's question about braid groups. There seems to be an interesting connection between the lower central series of subgroups of the pure braid groups and certain geometrical subgroups of braids that are described as being constrained by gropes. The latter subgroups (defined in the same way for general B_n and the pure braids) are given by the following recipe. We get a subgroup for each possible parenthization or partition of the strands and each selection of a sequence of grope types, one to encompass each subset of strands (possibly nested). The braid must respect the partition and the strands are not allowed to intersect the grope surfaces. It appears that the Brunnian braids and the more general subgroup described by Dr. Yetter are included in certain of these constrained subgroups. The only reference I know of is my own work--I would like to know if there are any studies of this kind of phenomena in existence. I am not yet sure of the full relationship, but examples and a more complete explanation are available from my research site: http://www.math.vt.edu/people/sforcey/class_home/research.htm or directly from: http://www.math.vt.edu/people/sforcey/gropebraid.htm Thanks for any and all input, Stefan Forcey Tennessee State University (currently at Virginia Tech) > From: "Hughes, James " > > 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