Bloch eigenmodes of PT-symmetric three-band topological insulators braid in sequences fixed by quaternion invariants; the effect is realized in coupled acoustic cavities with a synthetic dimension.
The Fundamental Group of $SO(n)$ Via Quotients of Braid Groups
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abstract
We describe an algebraic proof of the well-known topological fact that $\pi_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional relation and turns out to be a nontrivial central extension by $Z/2Z$ of the corresponding group of rotational symmetries of the hyperoctahedron in dimension $n$.
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Topological Braiding of Bloch Eigenmodes Protected by Non-Abelian Quaternion Invariants
Bloch eigenmodes of PT-symmetric three-band topological insulators braid in sequences fixed by quaternion invariants; the effect is realized in coupled acoustic cavities with a synthetic dimension.