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 SU q (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.

Braided fermions from Hurwitz algebras

“…Care must be taken to account for the permutations induced by braidings when systematically exchanging braiding for twisting [24]. It can likewise be shown that any orientable ribbon braid may be manipulated such that all the twisting of the ribbons is removed, exchanged for braiding [25]. One may write this as [0, 0, 0]B = B[0, 0, 0] where B ∈ B c 3 .…”