Knot (mathematics) - Generalization

Generalization

In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold with a submanifold, one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to . Traditional knots form the case where and or .

The Schoenflies theorem states that the circle does not knot in the 2-sphere—every circle in the 2-sphere is isotopic to the standard circle. Alexander's theorem states that the 2-sphere does not smoothly (or PL or tame topologically) knot in the 3-sphere. In the tame topological category, it's known that the -sphere does not knot in the -sphere for all . This is a theorem of Brown and Mazur. The Alexander horned sphere is an example of a knotted 2-sphere in the 3-sphere which is not tame. In the smooth category, the -sphere is known not to knot in the -sphere provided . The case is a long-outstanding problem closely related to the question: does the 4-ball admit an exotic smooth structure?

Haefliger proved that there are no smooth j-dimensional knots in provided, and gave further examples of knotted spheres for all such that . is called the codimension of the knot. An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of in form a group, with group operation given by the connect sum, provided the co-dimension is greater than two. Haefliger based his work on Smale's h-cobordism theorem. One of Smale's theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory. If one allows topological or PL-isotopies, Zeeman proved that spheres do not knot when the co-dimension is larger than two. See a generalization to manifolds.