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.)

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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.)
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Dan Asimov
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NASA Ames Research Center
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