# Braid Groups (Graduate Texts in Mathematics)

## Christian Kassel

Language: English

Pages: 338

ISBN: 1441922202

Format: PDF / Kindle (mobi) / ePub

The authors introduce the basic theory of braid groups, highlighting several definitions showing their equivalence. This is followed by a treatment of the relationship between braids, knots and links. Important results then look at linearity and orderability.

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arbitrary point d ∈ ∂ Σ lying over d and consider the relative homology group H = H1 (Σ, Gd; Z), where Gd is the G-orbit of d, i.e., the set of all points of Σ lying over d. The action of G on Σ induces a left action of G on H and turns H into a left module over the group ring Z[G]. This module is free of rank n = rk H1 (Σ; Z). This follows from the fact that Σ deformation retracts onto a union of n simple closed loops on Σ meeting only at their common origin d (here we crucially use the

⊂ R3 form a Conway triple if they coincide outside a 3-ball in R3 and look as in Figure 3.4 inside this ball. The Alexander–Conway polynomial of links is a mapping ∇ assigning to every oriented link L ⊂ R3 a Laurent polynomial ∇(L) ∈ Z[s, s−1 ] satisfying the following three axioms: (i) ∇(L) is invariant under isotopy of L; (ii) if L is a trivial knot, then ∇(L) = 1; (iii) for any Conway triple L+ , L− , L0 ⊂ R3 , ∇(L+ ) − ∇(L− ) = (s−1 − s) ∇(L0 ) . The latter equality is known as the

(i, m + 2) cancel each other. Similarly, for any i = 1, . . . , m, the loops ξm+1,i and ξm+2,i are homotopic and the contributions of the pairs (m + 1, i), (m + 2, i) cancel each other. Therefore N, α is preserved under the move. N α− zm−+1 α zm−+2 zm+1 zm+2 α− α Fig. 3.9. Additional crossings We say that a spanning arc α on (D, Q) can be isotopped oﬀ a noodle N if there is a continuous family of spanning arcs {αs }s∈[0,1] on (D, Q) such that α0 = α and α1 is disjoint from N . Such a

basic properties of Sn . 4.1 The symmetric groups The symmetric group Sn with n ≥ 1 is the group of all permutations of the set {1, 2, . . . , n}. The group law of Sn is the composition of permutations, and the neutral element is the identity permutation that ﬁxes all elements of {1, 2, . . . , n}. 4.1.1 A presentation of Sn by generators and relations Fix an integer n ≥ 1. For integers i, j such that 1 ≤ i < j ≤ n, we denote by τi,j the permutation exchanging i and j and leaving the other

We saw in the proof of Lemma 4.7 (a) that if t1 , . . . , tr are the transpositions deﬁned by (4.7), then I(w) = {t1 , . . . , tr }. If λ(wsj ) < λ(w), then sj ∈ I(w) by Lemma 4.7 (c). Therefore, sj = tk for some k ∈ {1, . . . , r}. By (4.8), wsj = wtk = si1 · · · sik · · · sir . The second claim is deduced from the ﬁrst one by replacing w with w−1 . Corollary 4.9. Let w ∈ Sn . If λ(wsj ) < λ(w) for some j ∈ {1, . . . , n − 1}, then there is a reduced expression for w ending with sj . If λ(sj w)