Alternating normal forms for braids and locally Garside monoids

Abstract

Communicated by C. Kassel MSC: 20F36 20M05 06F05 a b s t r a c tWe describe new types of normal forms for braid monoids, Artin-Tits monoids, and, more generally, for all monoids in which divisibility has some convenient lattice properties ("locally Garside monoids"). We show that, in the case of braids, one of these normal forms coincides with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the standard order ("Dehornoy order") of B n in terms of that of B n−1 , and to a quadratic upper bound for the complexity of this order.The first aim of this paper is to improve our understanding of well-order of positive braids and of the Burckel normal form of [5,6], which after more than ten years remain mysterious objects. This aim is achieved, at least partially, by giving a new, alternative definition for the Burckel normal form that makes it natural and easily computable. This new description is direct, involving right divisors only, while Burckel's original approach resorts to iterating some tricky reduction procedure. It turns out that the construction we describe below relies on a very general scheme for which many monoids are eligible, and we hope for further applications beyond the case of braids.After the seminal work of Garside [22], braid monoids are known to be equipped with a normal form, namely the so-called greedy normal form of [4,1,19,32], which gives for each element of the monoid a distinguished representative word. This normal form, which also exists in spherical Artin-Tits monoids and in Garside monoids that generalize them, is excellent both in theory and in practice as it provides a bi-automatic structure and it is easily computable [20,9,13].In this paper we proceed to construct a new type of normal form for braid monoids and their generalizations. Our construction keeps one of the ingredients of the (right) greedy normal form, namely considering the maximal right divisor that lies in some subset A, but, instead of taking for A the set of so-called simple elements, i.e., the divisors of the Garside element ∆, we choose A to be some standard parabolic submonoid M I of M, i.e., the monoid generated by some subset I of the standard generating set S. When I is a proper subset of S, the submonoid M I is a proper subset of M, and the construction stops after one step. However, by considering two parabolic submonoids M I , M J which together generate M, we can obtain a well-defined, unique decomposition consisting of alternating factors in M I and M J , as in the case of an amalgamated product. By considering convenient families of submonoids, we can iterate the process and obtain a unique normal form for each element of M. When it exists, typically in all Artin-Tits monoids, such a normal form is exactly as easy to compute as the greedy normal form, and, as the greedy form, it solves the word problem in quadratic time.The above construction is quite general, as it only requires the ground monoid M to be what is ...

Every braid admits a short sigma-definite expression

“…The rotating normal form is reminiscent of the alternating normal form introduced in [10] for the case of the monoid B + n -which is itself connected with Burckel's approach of [5]. It is also closely connected with the normal forms introduced in [17], which are other developments, in a different direction, of the alternating normal form.…”

“…It is also closely connected with the normal forms introduced in [17], which are other developments, in a different direction, of the alternating normal form. As the properties of B + * n and φ n are essentially the same as those of B + n and n , adapting the results of [10] is easy, and therefore constructing the rotating normal form is not very hardwhat will be harder is identifying the needed properties of rotating normal words, as will be done in subsequent sections.…”

“…The basic observation of [10] is that each braid in the monoid B As the word a 1,2 a 1,3 is alone in its equivalence class, the braid it represents cannot be right-divisible by a 1,2 . Therefore, the B + * 2 -tail of δ 2 3 is a 2 1,2 .…”

“…We prove Theorems 1 and 2 using the dual braid monoid B + * n associated with the BirmanKo-Lee generators and introducing a new normal form on B + * n , called the rotating normal form, which is analogous to the alternating normal form of [5] and [10]. The rotating normal form is based on the φ n -splitting operation, a natural way of expressing every n-strand dual braid in terms of a finite sequence of (n − 1)-strand dual braids.…”

Laver’s results and low-dimensional topology

“…In the general case, only partial results are known: for instance, it is shown in [28] that the family of all alternating normal n-strand braid words is recognized by a finite state automaton and, in [11], S. Burckel describes a recursive procedure for determining the rank in B + n . We conclude with extensions of the previous results involving other submonoids of the braid groups.…”