Non‐recursive functions, knots “with thick ropes,” and self‐clenching “thick” hyperspheres

Source abstract

Abstract We introduce an approach to certain geometric variational problems based on the use of the algorithmic unrecognizability of the n ‐dimensional sphere for n ≥ 5. Sometimes this approach allows one to prove the existence of infinitely many solutions of a considered variational problem. This recursion‐theoretic approach is applied in this paper to a class of functionals on the space of C 1.1 ‐smooth hypersurfaces diffeomorphic to S n in R n +1 , where n is any fixed number ≥ 5. The simplest of these functionals k v is defined by the formula k v (Σ n ) = ( vol (Σ n )) 1/ n / r (Σ n ), where r (Σ n ) denotes the radius of injectivity of the normal exponential map for Σ n ⊂ R n +l. We prove the existence of an infinite set of distinct locally minimal values of k v on the space of C 1.1 ‐smooth topological hyperspheres in R n +1 for any n ≥ 5. The functional k v naturally arises when one attempts to generalize knot theory in order to deal with embeddings and isotopies of “thick” circles and, more generally, “thick” spheres into Euclidean spaces. We introduce the notion of knot “with thick rope” types. The theory of knot “with thick rope” types turns out to be quite different from the classical knot theory because of the following result: There exists an infinite set of non‐trivial knot “with thick rope” types in codimension one for every dimension greater than or equal to five.