We study the topological structures in chiral nematic systems which exhibit biaxial behavior. Because of such biaxiality, the system is imbued with a set topological defect lines that inherit the algebraic properties of the quaternions from their non-Abelian fundamental group. This in turn provides topological stability to linked and braided structures formed out of these topological line defects. We give an overview of the complex structures that can be formed from linking non-Abelian defect lines and provide a framework in which they can be described mathematically as a colored braid theory. Such structures remain topologically stable to fluctuations and can serve as building blocks for larger structures such as networks and lattices of topologically linked defects. Additionally, we provide examples of experimental realizations of simple tangled structures in chiral nematic systems.