Vol. 42, No. 1, 1972

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On elementary ideals of polyhedra in the 3-sphere

Shin’ichi Kinoshita

Vol. 42 (1972), No. 1, 89–98

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,Sn2 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.

Mathematical Subject Classification
Primary: 55A25
Received: 4 May 1970
Revised: 13 January 1972
Published: 1 July 1972
Shin’ichi Kinoshita