From Christoffel words to braids

C. Kassel, Strasbourg

Abstract: In 1875 Christoffel published a paper in Latin (sic!) in which he associated a word w on a two-letter alphabet to the sequence formed by the multiples of an integer a considered modulo a coprime integer b. He then uses the word w to recover the expansion of a/b as a continued fraction. This may be one of the first systematic uses of words in mathematics. In this talk we'll start from Christoffel words and discuss various aspects of an interesting class of substitutions, the so-called Sturmian morphisms. In particular we'll show how Sturmian morphisms can be naturally identified with braids on four strings. This talk is based on joint work with C. Reutenauer, UQAM, Montreal.