Action of Braids on Self-distributive Systems

This paper is a survey of recent work about the action of braids on selfdistributive systems. We show how the braid word reversing technique allows one to use new self-distributive systems, leading in particular to a natural linear ordering of the braids. AMS Subject Classification: 20F36, 20N02. It has been observed for many years that there exist a connection between braids and left selfdistributive systems (LD-systems for short), defined as those algebraic systems consisting of a set equipped with a binary operation ∗ that satisfies the left self-distributivity identity x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z). (LD) In particular, D. Joyce [14] and S. Matveev [17], in independent works, have associated with every knot a particular LD-system that characterizes the isotopy type of the knot, and several variants of this approach have been subsequently proposed, with different names. Here, we do not try to associate with a given braid or knot a particular LD-system that gives information about it, but we fix an LD-system, and try to use it to get information about arbitrary braids. To this end, we can use the intuition of braid colouring: assuming that (S, ∗) is an LD-system, we use the elements of S as colours that we put on the strands of the braids, with the rule that a ∗ b is the new colour obtained when a strand of colour a overcrosses a strand of colour b. The left self-distributivity identity arises naturally as the compatibility condition needed for the colouring to be invariant under positive braid isotopy. In this way, we obtain for every LD-system S a well-defined action on S of the monoid B n of n-strand positive braids. In order to define an action of the whole group Bn, we must assume that the LD-system S has the additional property that all left translations are bijective, i.e., left division is always possible with a unique well-defined result. Such particular LD-systems have been called automorphic sets by E. Brieskorn [2], or racks by R. Fenn and C. Rourke [12]. So, in this way, we obtain an action of Bn on the n-th power of every automorphic set. Considering the known examples of automorphic sets leads to several classical representations of the braid groups, in particular Artin’s representation in the automorphisms of a free group and Burau representation. Automorphic sets are LD-systems of a very special type, in particular they are idempotent, or close to. In the recent years, new examples of LD-systems have appeared, in connection with results of set theory involving some strange LD-system [7]. These new examples are quite different from automorphic sets, and the question arises of extending the existence of a braid action to them. The aim of this paper is to explain how this can be done, at the expense of replacing an everywhere defined action with a partial action, in the case of a left cancellative LD-system, i.e., when we assume that the left translations are injective, but not necessarily surjective. Such a result makes most of the new examples of LD-systems eligible for a braid action. In particular, considering the action in the case of a certain left