The post here is an example of @oros terminology – the l(v) and s(v) can be bulkier than k, x, n, or m, but suddenly a w shows up (substring of v), and we can have both s(v) and s(w) without confusion.
I also see a_0 used as the smallest member of the cycle. (I suppose for both integer and rational cycles.)
That’s handy.
Maybe v_0 should be the parity vector corresponding to that smallest member, ie, the rotation of v with minimal \beta.
Then a_0 = \cfrac{\beta(v_0)}{2^{l(v_0)}-3^{s(v_0)}}.
Well, I guess a is a sequence of cycle members, while v is just a vector, rather than a sequence of rotations. Hmm, beats me. But lowest members and \beta-minimal vectors are things we talk about all the time.