On finite Thurston type orderings of braid groups

Details On finite Thurston type orderings of braid groups On finite Thurston type orderings of braid groups

2008-10-22

arxiv.org/abs/0810.4074v3

Groups Complex. Cryptol. 2 (2010) 123-155

Mathematics

Group Theory

25 pages, 2 figures; proof of Theorem 1 is corrected

Scientific paper

10.1515/GCC.2010.009

We prove that for any finite Thurston-type ordering $<_{T}$ on the braid group\ $B_{n}$, the restriction to the positive braid monoid $(B_{n}^{+},<_{T})$ is a\ well-ordered set of order type $\omega^{\omega^{n-2}}$. The proof uses a combi\ natorial description of the ordering $<_{T}$. Our combinatorial description is \ based on a new normal form for positive braids which we call the $\C$-normal fo\ rm. It can be seen as a generalization of Burckel's normal form and Dehornoy's \ $\Phi$-normal form (alternating normal form).

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