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

Abstract

AbstractWe 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 C1.1‐smooth hypersurfaces diffeomorphic to Sn in Rn+1, where n is any fixed number ≥ 5. The simplest of these functionals kv is defined by the formula kv(Σn) = (vol(Σn))1/n/r(Σn), where r(Σn) denotes the radius of injectivity of the normal exponential map for Σn ⊂ Rn+l. We prove the existence of an infinite set of distinct locally minimal values of kv on the space of C1.1‐smooth topological hyperspheres in Rn+1 for any n ≥ 5.The functional kv 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.