There exist conjugate simple braids whose associated permutations are not strongly conjugate

If two permutations are strongly conjugate, then their corresponding positive permutation braids (also called simple braids) are conjugate. In this paper we exhibit two conjugate simple braids whose associated permutations are not strongly conjugate. In terms of the grey and black graphs with vertices in the ultra summit set defined in [1], this result can be reformulated by saying that there are ultra summit sets with simple braids (hence with canonical length k = 1) in which the grey graph is not connected. Birman, Gebhardt and González Meneses have given similar examples, but with k ≥ 2 [1]. Recall that the set of simple braids on n strings is a basis of the Hecke algebra Hn. If two simple braids on n strings are conjugate, the associated permutations are centrally conjugate, which means that the coefficients of any central element of Hn corresponding to these simple braids are equal. Working on topological dynamics, Hall and Carvalho [8] have discovered two braids on 12 strings (one and its reverse) which are not conjugate. Since one braid is the reverse of the other, their corresponding permutations are centrally conjugate. For n ≤ 6 we checked that central conjugacy implies conjugacy of the corresponding simple braids.