Example 1.4. A simple, yet structurally very important example of the use of the diamond lemma is the identity problem and the conjugacy problem in the free group. Given a set of letters (alphabet) a1, . . . , an, we consider all possible words in this alphabet. Two words w, w′ are in the relation w → w′ if w′ is obtained from w by contraction of two successive letters aa−1 or a−1 a. These transformations are called elementary contractions. The Noetherian property is evident.
Fig. 1.4Proof of the Diamond lemma
How one can check the diamond lemma in this case? Let w be a word which admits two elementary contractions w → w′ and w → w″. If the contracted letters don’t intersect, then we can contract them all in any order and get a word w′″ which is a descendant for w′ and w″. If the letters intersect then we have a subword aa−1a or a−1aa−1 in w which is contracted by two possible ways. In this case the words w′ and w′ coincide.
Analogously, the conjugacy problem in the free group can be solved. For a word w = w1 . . . wk of length k, we consider words w(j) = w1+jw2+j . . . wk+j obtained from w by cyclic permutations 1 → 1 + j, . . . , k → k + j, where the sums are modulo k. It is clear that all these word are conjugate to w. The word w is called cyclically contractible if some of the words wj are contractible. We introduce the following relation on the cyclic words: w → w′ if there exist representatives w and w′ of the words w, w′ such that w → w′.
Evidently, two cyclic words w, w′ are conjugated if and only if there is a sequence of cyclic words w = w0 → w1. . . wp = w′ such that for any successive words either wj → wj+1 or wj+1 → wj. Thus, the conjugacy is the equivalence generated by the elementary relation →.
For cyclic words of length ≥ 3 the conditions of the diamond lemma can be checked in the same manner as for ordinary words. The verification of the conditions for words of length 2 is obvious.
Remark 1.4. As we could see in the examples considered above, two word contractions a → b, b → c can be independent if the contracted letters of one contraction are not involved in the other, and dependent if the both contractions use common letters. In the first case the commutative diagram of the diamond lemma can be constructed in one step: the word d is obtained from the word c “in the same way” as the word b is obtained from the word a, meanwhile the word d is obtained from the word b “in the same way” as the word c is obtained from the word a. Thus, we have a → b → d and a → c → d.
Such cases of “independent contractions” are connected usually to the phenomenon of “far commutativity”.
As for the case of dependent contractions, in the simplest situation dependence of the contractions a → b and a → c implies b ≡ c where ≡ is the identity relation. In the general case application of the diamond lemma to dependent contractions is much more difficult.
Chapter 2
Braid Theory
The present chapter is an introduction to the theory of braids, which has some nice intrinsically interesting properties. Here we highlight some ideas and definitions which will be important for our purposes later. In our overview of the basics of braid theory we will closely follow [Manturov, 2018].
2.1Definitions of the braid group
Below, we are going to give some definitions of the braid groups and to discuss some of their properties.
First we consider a geometrical definition of the braid group. Consider the lines {y = 0, z = 1} and {y = 0, z = 0} in
Definition 2.1. An m-strand braid is a set of m non-intersecting smooth paths connecting the chosen points on the first line to the points on the second line (in arbitrary order), such that the projection of each of these paths to Oz represents a diffeomorphism.
These smooth paths are called strands of the braid.
An example of a braid is shown in Fig. 2.1. It is natural to consider braids up to isotopy in
Definition 2.2. Two braids B0 and B1 are equal if they are isotopic; i.e., if there exists a continuous family Bt, t ∈ [0, 1] of braids starting at B0 and finishing at B1.
Definition 2.3. The set of (isotopy classes of) m-strand braids generates a group. The operation in this group is just juxtaposing one braid under the other and rescaling the z-coordinate.
Fig. 2.1A braid
The unit element or the unity of this group is the braid represented by all vertical parallel strands. The inverse element for a given braid is just its mirror image; see Fig. 2.2.
Fig. 2.2Unity. Operations in the braid group
Definition 2.4. The Artin m-strand braid group1 is the group of braids with the operation defined above. The braid group is denoted by Br(m).
The second definition of the braid group considered in the present chapter is an algebraic definition.
Definition 2.5. The m-strand braid group is the group given by the presentation with (m − 1) generators σ1, . . . , σm−1 and the following relations
for |i − j| ≥ 2 and
for 1 ≤ i ≤ m − 2.
These relations are called Artin’s relations.
Definition 2.6. Words in the alphabet of σ’s and σ−1’s will be referred to as braid words.