Superstrings, Knots, and Noncommutative Geometry in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$E^{{\text{(}}\infty {\text{)}}} $$ \end{document} Space

Within a general theory, a probabilisticjustification for a compactification which reduces aninfinite-dimensional spacetime \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$E^{{\text{(}}\infty {\text{)}}} (n = \infty )$$ \end{document} to afour-dimensional one (DT = n = 4) isproposed. The effective Hausdorff dimension of this spaceis is given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\langle \dim _{\text{H}} E^{{\text{(}}\infty {\text{)}}} \rangle = d_{\text{H}} = 4 + \Phi ^3 ,{\text{ where }}\Phi ^3 = 1/[4 + \Phi ^3 ]$$ \end{document} is a PV number and φ = (√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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$E^{{\text{(}}\infty {\text{)}}} $$ \end{document}, statistical mechanics, and Jones polynomials are drawn.This allows a better insight into the nature of theproposed compactification, the associated \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$E^{{\text{(}}\infty {\text{)}}} $$ \end{document} space, and thePisot–Vijayvaraghavan number 1/φ3= 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.