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

Christian Kassel·Christophe Reutenauer
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Abstract

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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Christian Kassel, Christophe Reutenauer. 2005-07-11. Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2. https://doi.org/10.1007/s10231-006-0008-z

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