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On finite Thurston-type orderings of braid groups. (English) Zbl 1241.20040

Summary: 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 combinatorial description of the ordering \(<_T\). Our combinatorial description is based on a new normal form for positive braids which we call the \(\mathcal C\)-normal form. It can be seen as a generalization of Burckel’s normal form and Dehornoy’s \(\Phi\)-normal form (alternating normal form).

MSC:

20F36 Braid groups; Artin groups
20F60 Ordered groups (group-theoretic aspects)
06F15 Ordered groups
20F05 Generators, relations, and presentations of groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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