Note on the Penrose-Mackay tiling, Hirzbruch signature of M4 and knots in E(∞) space

Abstract

Mackay's generalization of Penrose tiling is shown to be related to the Hirzbruch signature, δ, of four manifolds in the case of a Murray-von Neumann continuous dimension in E(∞) space. It is further shown that σ is numerically identical to the Jones knot invariant, VL, for the right-handed trefoil. Finally, a conjecture regarding the theorem of local isomorphism is presented.