# Why use class multiplication to describe topological entangling and merging?

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I'm studying some references about topological defects in ordered media like *Soft matter physics: An introduction* by Kleman and the Review modern physics paper *The topological theory of defects in ordered media* by Mermin. In both of them, the authors emphasize the class multiplication instead of elements multiplication for describing disclination merging and entangling, and then admit the arbitrariness of the class multiplication. However, I have some problem to understand this. E.g., for a bi-axial liquid crystal, disclinations are classified by $\pi_1(SO(3)/D_2) = Q_8$. This is a quaternion group who has five conjugacy classes $\{1\}, \{-1\},\{i,-i\},\{j,-j\},\{k,-k\}$. I understand the fact that elements $i$ and $-i$ describing , e.g., the disclination and anti-disclination in $yz$ plane, so it is reasonable to group them together. However, when I do defects merging or entangling using the class multiplication, as suggest in the references, I would have problem to predict the result: $\{i,-i\}$ mutiplities $\{i,-i\}$ can either give me $\{1\}$ or $\{-1\}$ who are different defects. Why doesn't one use the elements multiplication directly which doesn't lead to the arbitrary? Put differently, why is it necessary to use class multiplication? Can any one give me any hints?