Knotted boundaries and braid only form of braided belts

Abstract

The Helon model identifies Standard Model quarks and leptons with certain framed braids joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts (or simply belts). Twisting and braiding of ribbons composing braided 3-belts are interchangeable, and it was shown in the literature that any braided 3-belt can be written in a pure twist form, specified by a vector of three multiples of half integers [a,b,c], a topological invariant. This paper identifies the set of braided 3-belts that can be written in a braid only form in which all twisting is eliminated. For these braids an algorithm to calculate the braid word is determined which allows the braid only word of every braided 3-belt to be written in a canonical form. It is furthermore demonstrated that the set of braided 3-belts do not form a group, due to a lack of isogeny. The conditions under which the boundary of a braided 3-belt is a knot are determined, and a formula for the Jones polynomial for knotted boundaries is derived. Considering knotted boundaries makes it possible to relate the Helon model to a model of quarks and leptons in terms of quantum trefoil knots, understood as representation of the quantum group SUq(2). Associating representations of a quantum group to the boundary of braided belts provides a possible means of developing the gauge symmetries of interacting braided belts in future work.