Superstrings, Knots, and Noncommutative Geometry in E^{{text{(}}infty {text{)}}} Space

Within a general theory, a probabilisticjustification for a compactification which reduces aninfinite-dimensional spacetime to afour-dimensional one (D_T = n = 4) isproposed. The effective Hausdorff dimension of this spaceis is given by is a PV number and = (5– 1)/2 is the golden mean. The derivation makes use of various results from knot theory,four-manifolds, noncommutative geometry, quasiperiodictiling, and Fredholm operators. In addition somerelevant analogies between , statistical mechanics, and Jones polynomials are drawn.This allows a better insight into the nature of theproposed compactification, the associated space, and thePisot–Vijayvaraghavan number 1/³= 4.236067977 representing its dimension. This dimensionis in turn shown to be capable of a naturalinterpretation in terms of the Jones knot invariant andthe signature of four-manifolds. This brings the work near to the context of Witten andDonaldson topological quantum field theory.