Base points

Let \(z=(\omega,\omega^2,\dots,\omega^n=1)\) and \(z'=(\omega^p,\omega^{2p},\dots,1)\) (where \(p,n\) are coprime - \(p\) should be \(\ell\))

How are braids at \(z\) and at \(z'\) related?

What is a natural braid from \(z\) to \(z'\)?

Note that \(z'=\sigma\cdot z\) for some \(\sigma\in S_n\). Given \(p\), we should be able to write down \(\sigma\) and then a braid.

Following on from this, we need to write down the braids for \(D(n,\infty/\ell)\) circluar choreographies, and others of type \(D(n,k/\ell)\).

Homology

Homology in \(P_n\) = winding numbers \(w_{ij}\).

How is the symmetry group built in to this?

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