Fast algorithmic Nielsen-Thurston classification of four-strand braids

Matthieu Calvez (1) , Bert Wiest (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We give an algorithm which decides the Nielsen-Thurston type of a given four-strand braid. The complexity of our algorithm is quadratic with respect to word length. The proof of its validity is based on a result which states that for a reducible 4-braid which is as short as possible within its conjugacy class (short in the sense of Garside), reducing curves surrounding three punctures must be round or almost round.

Mots clés: braids; Nielsen-Thurston classification; algorithm; Garside theory

Domaines: Théorie des groupes [math.GR]

Matthieu Calvez, Bert Wiest. Fast algorithmic Nielsen-Thurston classification of four-strand braids. Journal of Knot Theory and Its Ramifications, 2012, 21 (5), ⟨10.1142/S0218216511009959⟩. ⟨hal-00468432v4⟩