From: asimov@nas.nasa.gov (Daniel A. Asimov) Newsgroups: sci.math.research Subject: Arrangements of Geometric k-Spheres in R^(2k+1)- replacement article Date: Thu, 12 Jan 1995 14:49:52 -0800 (This is a generalization of a problem I posed around two years ago.) Throughout this article, let n = 2k+1, where k is some fixed integer >= 0; Define a k-hoop in R^n to be any geometric k-sphere of radius 1 in R^n. Denote the space of all k-hoops in R^n by H(k). The space H(k) is endowed with a natural topology (as the product of R^n and the Grassmannian G(k,n) of k-planes in R^n). Define an r-hoople to be a set of r disjoint k-hoops. Denote the space of all such r-hooples by H(k;r). This space H(k;r) also has a natural topology on it (from the deleted rth Cartesian power of H(k), which is then factored out by the symmetric group S(r)). * * * QUESTION: How many arcwise-connected components does H(k:r) have, and what is a representative r-hoople for each one? * * * For a simple case, what is the answer for k = 2, r = 3 ??? (I don't know.) ------------------------------------------------------------------------------ Corrected version of this section of the article: The original question considered the case k = 1, r = 3, and was answered empirically rather than with a proof. The apparent answer here is that there are exactly 6 components, represented by the following cases: (The word "hoop" is used here to mean a 1-hoop. In R^3, of course.) a) three mutually unlinked hoops, b) two linked hoops and the third far away, c) a chain of three hoops, d,e) a circular chain of mutually linked hoops (two mirror-image versions) f) left to the reader as a puzzle. Apparently, each circular chain of three hoops lies in one of two components of H(1;3) that represent mirror-images of each other (and there is a theorem that the Borromean rings cannot be represented by round circles). The case k = 1, r = 4 has been studied by Stein Kulseth of Norway, who came up with the empirical result of 33 distinct components (last I heard). (Apologies to those who were misled by my almost certainly erroneous assertion in the original posting that the circular chain of 3 hoops is amphicheiral. Empirically it seems clear that this can occur in two mirror-image versions.) ------------------------------------------------------------------------------ Dan Asimov Mail Stop T27A-1 NASA Ames Research Center Moffett Field, CA 94035-1000 asimov@nas.nasa.gov (415) 604-4799 w (415) 604-3957 fax