Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2

Details Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2 Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2

2005-07-11

arxiv.org/abs/math/0507219v1

Ann. Mat. Pura Appl. 186 (2007), 317-339

Mathematics

Group Theory

25 pages, 4 figures

Scientific paper

10.1007/s10231-006-0008-z

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F_2) of automorphisms of the rank two free group F_2 and show that it can be realized as a monoid in the group B_4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F_2 lifting any given basis of the free abelian group Z^2. We further give an algorithm allowing to decide whether two elements of F_2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.

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