Linking is a fascinating phenomenon in topology, but the only chance of proving anything about it will require the use of a definition of linking. --Dan Fred wrote: << Paraphrased and extended from exercises 5.5--5.10 of Ian R. Porteous "Clifford Algebras and the Classical Groups" Cambridge (2000) --- surely familiar to those with a background in topology, but new to me. The circles and spheres are standard, rather than just homeomorphs. << Given disjoint circles C,D in 3-space, consider the (Gauss) map from vectors joining pairs of points, X on C and Y on D, to the unit sphere: (X, Y) -> (X - Y)/|X - Y|. Show that this map is surjective if and only if the circles are linked; and specify its fibres (pre-images). Show that two disjoint great circles in spherical 3-space are always linked. Specify precisely a criterion to decide whether two circles in 3-space are (not) linked. Design a simple, effective algorithm to decide whether two circles in 3-space are linked. Show that two spheres may be linked in 5-space, but not in 4-space or 6-space. Generalise (inductively?) to decide whether k-sphere and l-sphere in (k+l+1)- space are linked
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On 10/20/09, Dan Asimov <dasimov@earthlink.net> wrote:
Linking is a fascinating phenomenon in topology, but the only chance of proving anything about it will require the use of a definition of linking.
Which is not supplied; although at a later stage (ex 5.8) a definition is requested! It turns out that for standard k-spheres, a simple and intuitive criterion is available. However, as far as I can see, an algorithm (also quite straightforward) for deciding whether spheres are linked is related neither to this criterion, nor to the Gaussian map property, nor to any more topologically motivated definition of linking. WFL
Fred wrote:
<< Paraphrased and extended from exercises 5.5--5.10 of Ian R. Porteous "Clifford Algebras and the Classical Groups" Cambridge (2000) --- surely familiar to those with a background in topology, but new to me. The circles and spheres are standard, rather than just homeomorphs.
<< Given disjoint circles C,D in 3-space, consider the (Gauss) map from vectors joining pairs of points, X on C and Y on D, to the unit sphere: (X, Y) -> (X - Y)/|X - Y|.
Show that this map is surjective if and only if the circles are linked; and specify its fibres (pre-images).
Show that two disjoint great circles in spherical 3-space are always linked.
Specify precisely a criterion to decide whether two circles in 3-space are (not) linked.
Design a simple, effective algorithm to decide whether two circles in 3-space are linked.
Show that two spheres may be linked in 5-space, but not in 4-space or 6-space.
Generalise (inductively?) to decide whether k-sphere and l-sphere in (k+l+1)- space are linked
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Fred lunnon