Narrow Results

This is the next step of uncovering the relation between string theory and exotic smooth ℝ⁴. Exotic smoothness of ℝ⁴ is correlated with D6-brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum Dp-branes as well to KK theory. These branes emerge when there are nontrivial NS–NS H-fluxes on S³ where the topological classes are determined by wild embeddings S²→S³. Then wild embeddings of higher dimensional p-complexes into Sⁿ correspond to Dp-branes. These wild embeddings as constructed by using gropes are basic objects to understand exotic smoothness as well Casson handles. Next we build C^⋆-algebras corresponding to the embeddings. Finally we consider topological quantum D-branes as those which emerge from wild embeddings in question. We construct an action for these quantum D-branes and show that the classical limit agrees with the Born–Infeld action such that flat branes = usual embeddings.

Bing, Bothe and Shilepsky studied knots that were wild at every point, and Bothe developed an invariant for a certain subclass of these knots. This paper develops invariants which distinguish ambient isotopy equivalence classes of these knots (which include the "Bing sling") and shows that there is an uncountable number of inequivalent of Bing sling type knots. As an aside, it is also shown that it is possible for inequivalent wild knots to have homeomorphic complements.

We note a simple algebraic proof of Frolkina’s result that ${ℝ}^3$ does not contain uncountably many pairwise disjoint copies of the Möbius band, and of a similar result in higher dimensions.

The paper constructs an example of a (non-ambient) isotopy of a (topological) knot in ${ℝ}^3$ which does not extend to an isotopy of a two-component link whose second component is a meridianal ring of this knot. By constructing the example, the paper answers a question raised in 2005 by Melikhov. Apart from presenting the corresponding example, the paper will discuss the consequences that the discovery of this example has for attacking a problem that was in literature asked in 1974 by Rolfsen: Namely deciding, whether there exist knots (and it is clear that at most everywhere wild knots might have such a property) that even with respect to ordinary (not necessarily ambient) isotopy are non-equivalent to the trivial knot.

In this paper, we discuss wild embeddings like Alexanders horned ball and relate them to fractal spaces. We build a C*-algebra corresponding to a wild embedding. We argue that a wild embedding is the result of a quantization process applied to a tame embedding. Therefore, quantum states are directly the wild embeddings. Then we give an example of a wild embedding in the four-dimensional spacetime. We discuss the consequences for cosmology.