By Patrick Dehornoy
This is the award-winning monograph of the Sunyer i Balaguer Prize 1999. The ebook offers lately chanced on connections among Artin’s braid teams and left self-distributive structures, that are units outfitted with a binary operation gratifying the id x(yz) = (xy)(xz). even though no longer a accomplished direction, the exposition is self-contained, and plenty of easy effects are proven. particularly, the 1st chapters contain an intensive algebraic research of Artin’s braid groups.
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Extra resources for Braids and Self-Distributivity
Things become more interesting when we allow colours to change at crossings. Let us begin with the case of positive braid diagrams. According to our principle of propagating the colours downwards from the top of the braid diagram, it is natural to fix the rules of colouring so that the colours after a crossing (the ‘new’ colours) are functions of the colours before the crossings (the ‘old’ colours). Keeping the colours corresponds to the simplest function, namely identity. The next step is to assume that only one colour may change, say the colour of the front strand, and the new colour depends only on the two colours of the strands that have crossed.
The previous list more or less exhausts classical LD-quasigroups. A few more are described in exercises below, but they are unessential variations. 2: Braid Colourings that using braid colourings and resorting to various (classical) LD-quasigroups leads to several (classical) representation results for braid groups. 13. Can we find new examples of LD-quasigroups, and deduce further properties of braids? The answer to the first part is positive, but it seems that the answer to the second part is essentially negative.
An ) • w. 1) 14 Chapter I: Braids vs. , the colourings have to be invariant under braid relations. 2. Assume that (S, ∧) is a binary system. 2) if and only if (S, ∧) satisfies the left self-distributivity identity x ∧ (y ∧ z) = (x ∧ y) ∧ (x ∧ z). Proof. Compare the diagrams: a b c a a∧b a c a a ∧b a∧c a a∧(b∧c) (a∧b)∧(a∧c) a∧b (LD) b c a∧(b∧c) a∧b a, b b∧c a b a. The lower colours on the left strand coincide for every initial choice of a, b, c if and only if the operation ∧ satisfies Identity (LD).