Fast Algorithmic Nielsen–thurston Classification of Four-Strand Braids

Abstract

Abstract. 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. As an application, we give a polynomial time solution to the conjugacy problem for non-pseudo-Anosov four-strand braids.

Efficient Algorithm for Recognizing the Nielsen–Thurston Type of a Three–Strand Braid

“…[2]). Regarding pseudoAnosov and reducible braids, there are several recognition methods [1]- [7] using either traintracks or modifications of summit sets. In this paper.…”

“…Efficient (on the input word length) algorithms for recognizing the type of a given braid are described in [6,7], but these algorithms are also based on the use of summit sets (more exactly, supersummit sets). For these algorithms the operation time is estimated by O(l 2 ) in the case of four strands and O(l 3 ) in the case of an arbitrary number of strands, where l denotes the length of input word in the classical Artin generators σ i .…”

A fast solution to the conjugacy problem in the four-strand braid group

“…(1) Determine the dynamical types of x and y, using [9]. If they are not the same, answer "x and y are not conjugate" and STOP.…”

“…In [9], the authors answered this question in the case of the group B 4 : they produced an algorithm of complexity O(ℓ 2 ) to decide the Nielsen-Thurston type of any given 4-strand braid of length ℓ. Thus in the group B 4 , in order to solve CDP and CSP it is sufficient to solve these problems for pairs of elements which are known to be of the same dynamical type (as pairs of braids of different dynamical type are never conjugate).…”

“…The algorithm given in [9] also implies a solution to CDP and CSP for reducible four-strand braids of length at most ℓ in time O(ℓ 2 ). The main lemma here is that for braids with at most 3 strands and of length at most ℓ, the problems CDP and CSP are solvable in time O(ℓ 2 ), see [9].…”