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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