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Embedding Knots in Trivial Knots

Embedding Knots in Trivial Knots F. GONZALEZ-ACUNA AND JOSE MARIA MONTESINOS It is known (see [2]) that not every n-knot is doubly nullcobordant, i.e. a cross section of a trivial (n + l)-knot. Thus, in general, a knot cannot be embedded with codimension one in a trivial knot. Here we will show that every knot can be embedded with codimension two in a trivial knot. We work in the smooth category. All manifolds are oriented and all n + 2 (co)homology groups have Z coefficients. An n-sphere in S which bounds an + 2 (n + l)-ball in S" is said to be trivially embedded. n + 2 2 THEOREM. If S" is embedded in S and M"~ is a codimension 2, compact, oriented {not necessarily connected) submanifold of S" without boundary, then there is n+2 an n-sphere S" trivially embedded in S such that S" and S" intersect transversely and n 2 2 S"nS" = M ~ . Moreover,for every xeH {S"-M"- ), Lk(x,M) = Lk(i (x),S") 1 # n 2 2 where i^ : H {S -M"- ) -> H^S"* -^) is induced by inclusion. n n + 2 Proof. It is sufficient to consider the case where S is trivially embedded http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Embedding Knots in Trivial Knots

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/14.3.238
Publisher site
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Abstract

F. GONZALEZ-ACUNA AND JOSE MARIA MONTESINOS It is known (see [2]) that not every n-knot is doubly nullcobordant, i.e. a cross section of a trivial (n + l)-knot. Thus, in general, a knot cannot be embedded with codimension one in a trivial knot. Here we will show that every knot can be embedded with codimension two in a trivial knot. We work in the smooth category. All manifolds are oriented and all n + 2 (co)homology groups have Z coefficients. An n-sphere in S which bounds an + 2 (n + l)-ball in S" is said to be trivially embedded. n + 2 2 THEOREM. If S" is embedded in S and M"~ is a codimension 2, compact, oriented {not necessarily connected) submanifold of S" without boundary, then there is n+2 an n-sphere S" trivially embedded in S such that S" and S" intersect transversely and n 2 2 S"nS" = M ~ . Moreover,for every xeH {S"-M"- ), Lk(x,M) = Lk(i (x),S") 1 # n 2 2 where i^ : H {S -M"- ) -> H^S"* -^) is induced by inclusion. n n + 2 Proof. It is sufficient to consider the case where S is trivially embedded

Journal

Bulletin of the London Mathematical Society – Wiley

Published: May 1, 1982

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