Classifying Non-trivial Links of Biaxial Nematic Defect Lines

Unlike uniaxial nematics, biaxial nematic systems have a non-Abelian fundamental group. As a consequence, defects lines that form in these systems have non-trivial combination rules that obey the algebra of its fundamental group, the quaternions. This non-Abelian nature of the defect lines leads to non-trivial linking, in which two defects braided around each other may remain linked depending on the individual defect types, a feature that is not present in systems with Abelian properties. This allows one to form links of N defect lines that may not necessarily be equivalent to N unknots. We attempt to model this system as a colored braid theory subject to a set of equivalence relations on its generators in order to determine a classification system for the possible non-trivial links