Homotopy classification of knotted defects in ordered media

We give a homotopy classification of the global defects in ordered media and explain it via the example of biaxial nematic liquid crystals, that is, systems where the order parameter space is the quotient of the three-sphere <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> by the quaternion group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Q</mml:mi> </mml:math> . As our mathematical model, we consider continuous maps from complements of spatial graphs to the space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mi>/</mml:mi> <mml:mi>Q</mml:mi> </mml:mrow> </mml:math> modulo a certain equivalence relation and find that the equivalence classes are enumerated by the six subgroups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Q</mml:mi> </mml:math> . Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Q</mml:mi> </mml:math> ; once we pass to planar diagrams, these labels can be refined to elements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Q</mml:mi> </mml:math> associated with each arc. The same classification scheme applies not only in the case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>Q</mml:mi> </mml:math> but also to arbitrary groups.