Why use class multiplication to describe topological entangling and merging?

Question

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 emphasis the class multiplication instead of elements multiplication for describing disclination merge and entangle, 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, what is the necessary of using class multiplication? Can any one give me any hits?

This post imported from StackExchange Physics at 2015-02-11 11:56 (UTC), posted by SE-user hongchaniyi

Comments

"ordered media" means symmetry breaking. This question is related to symmetry breaking order which is not related to topological order and topological phase.

This post imported from StackExchange Physics at 2015-02-11 11:56 (UTC), posted by SE-user Xiao-Gang Wen