|
Let K be a polygonal simple
closed curve (a knot) in a 3-sphere S3. For each nonnegative integer d the d-th
elementary ideal Ed of K in the integral group-ring over an infinite cyclic group is
defined by R. H. Fox. The ideal Ed of K is a topological invariant of the position of
K in lS3. This method has been applied to various more general settings, for
instance, links in |S3,Sn−2 in Sn(n > 2) and etc. In this paper the d-th elementary
ideals Ed(l) are associated to each (n − 2)-dimensional integral cycle l on a
polyhedron L in an n-sphere Sn(n > 2) that does not separate Sn. The collection of
Ed(l) for all possible l on L forms a toplogical invariant of the position of L in
Sn.
In §2 we prove theorems of the d-th elementary ideal Ed(l) associated with an
(n − 2)-dimensional integral l on a polyhedron L in Sn that does not separate
Sn(n > 2). In §3 we will consider to the case of polyhedra in Sa. After studying an
example of a 𝜃-curve in Sa in §4, we reconsider knots and links from this point of
view in §5. In §6 we will give a remark on a Torres’ formula for a link in S3 ([7]) from
this point of view.
Our discussion will be based on Fox’s free differential calculus ([1], [2], [3]),
though other methods, especially the covering space technique, would also be helpful.
We need some minor adjustment of free differential calculus that will be given in
§1.
|
