Knotted boundaries and braid only form of braided belts

TL;DR

This paper explores the properties of braided 3-belts in the Helon model, identifying conditions for braid-only forms, deriving a canonical braid word, and relating knotted boundaries to quantum trefoil knots and gauge symmetries.

Contribution

It introduces an algorithm for canonical braid words, characterizes when boundaries are knots, and links braided belts to quantum group representations, advancing the topological understanding of the Helon model.

Findings

Certain braided 3-belts can be expressed in braid-only form with an explicit algorithm.

The set of braided 3-belts does not form a group due to lack of isogeny.

Formulas for the Jones polynomial of knotted boundaries are derived.

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.