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Rudimentary knots are invoked to generate a representation of the elementary particles, a model that endows the particles with visualizable structure. The model correlates with the basic tenets, taxonomy, and interactions of the Standard Model, but goes beyond it in a number of important ways, the most significant being that all particles (hadrons and leptons, fermions and bosons) and interactions share a common topology. Among other consequences of the modeling are the topological basis for isospin invariance and its connection to electric charge, the necessary identity of electron and proton charge magnitudes, and the existence of precisely three generations on the particle family tree. The salient feature of the model is that the elementary particles are viewed not as discrete, point-like objects in a vacuum but rather as sustainable, membrane-like distortions embodying curvature and torsion in and of an otherwise featureless continuum and that their manifest physical attributes correlate with the distortion. There are additional connections to the theories of fiber bundles, superstrings and instantons and, historically, to the work of Kelvin in the mid-nineteenth century and Cartan in the 1920s among others.

Apart from their generic relationship to knots and their application to particle physics [1], flattened Moebius strips (FMS) are of intrinsic interest as elements of a genus with specific rules of combination and a unique taxonomy. Here, FMS taxonomy is developed in detail from combinatorial and lexicographic points of view which include notions of degeneracy, completeness and excited states. The results are compared to the standard, spin-parameterized, abstract hierarchy derived by group-theoretic arguments as the direct product of vector spin spaces [2]. A review of the notion of excited states then leads to a new and different model of Beta decay that employs only fusion and fission. There is additional discussion of the relationship between twist and charge and an operator/tensor formulation of the fusion and fission of basic FMS units. Associating a Hopf algebra to FMS operations as a step toward a topological quantum field theory is also investigated. The notion of spinor/twistor networks is seen to emerge from a consideration of FMS configurations for higher values of twist and the introduction of a mode dual to the canonical FMS configuration. The last section discusses the connection of the MS genus to fiber bundle/gauge theory, the concept of spin, and the Dirac equation of the electron.

The taxonomy of flattened Moebius strips (FMS) is reexamined in order to systematize the basis for its development. An FMS is broadly characterized by its twist and its direction of traverse. All values of twist can be realized by combining elementary FMS configurations in a process called fusion but the result is degenerate; a multiplicity of configurations can exist with the same value of twist. The development of degeneracy is discussed in terms of several structural factors and two principles, conservation of twist and continuity of traverse. The principles implicate a corresponding pair of constructs, a process of symbolic convolution, and the inner product of symbolic vectors. Combining constructs and structural factors leads to a systematically developed taxonomy in terms of twist categories assembled from permutation groups. Taxonomical structure is also graphically revealed by the geometry of an expository edifice that validates the convolution process while displaying the products of fusion. A formulation that combines some of the algebraic precepts of Quantum Mechanics with the primitive combinatorics and degeneracies inherent to the FMS genus is developed. The potential for further investigation and application is also discussed. An appendix outlines the planar extension of the fusion concept and another summarizes a related application of convolution.