Index of Notation N (nonnegative integers) Z (integers) Q (rationals) R (reals) C (complex numbers) Introduction Bn (braid group), 1 B+ n (braid monoid), 1 B+∗ n (dual braid monoid), 1 Sn (symmetric group), 1 PBn (pure braid group), 1 Chapter I Bn, B∞ (braid group), 9 σ i (braid), 9 w (braid word), 9 (w) (length), 9 D2 (disk), 10 Dn (punctured disk), 12 MCG(S, P) (mapping class group), 12 Fn (free group), 13 Bn + (braid monoid), 14 (β) (length of a positive braid), 14 δn, ∆n (fundamental braids), 14 φn (conjugation by δn), 14 Φn (flip automorphism), 14 β β (left divisor), 16 DL(β) (left denominator), 17 NL(β) (left numerator), 17 DR(β) (right denominator), 17 NR(β) (right numerator), 17 Chapter II sh (shift endomorphism), 22 n , (σ-ordering), 21 A (Property), 21 C (Property), 21 S (Property), 28 Φ (σΦ-ordering), 24 w (equivalence class), 25 [Bn, Bn] (commutator subgroup), 33 ω (ordinal), 38 emin r (Φ3-normal form), 40 Chapter III RG (group algebra), 44 L2(G) (Hilbert space), 45 b (closed braid), 47 ω(β) (twist), 48 β{t} (braid game), 51 G3 (sequence of braids), 51 IΣk (logical system), 52 G∞ (sequence of braids), 52 deg(β) (degree of a braid), 53 WOf (combinatorial principle), 53 Ack, Ackr (Ackermann function), 53 Chapter IV LD (left self-distributivity law), 55 x • β (braid action), 56 β ∗ β (braid operation), 58 Q sh (β1, ..., βn) (shifted product), 58 Bsp (special braids), 59 C∞ (Property), 60 (iterated left divisor), 60 Tn (terms), 63 =LD (LD-equivalence), 63 ∂t (term), 65 x[k] (right power), 65 left(t) (left subterm), 65 Ai (Property), 73 t LD t (left subterm), 76 LDα (operator), 77 GLD (geometry monoid), 77 GLD (geometry monoid), 78 χt (blueprint), 79 [[t]] (blueprint), 79 Fn LD (free LD-system), 82 F LDM n (free LD-monoid), 82 Chapter V a, b, ... , A, B, ... (braids), 89 redw (handle reduction), 90 Div(β) (set of left divisors), 93 h(w) (number of handles), 97 π(w) (main prefix), 97 e(w) (sign of main prefix), 97 c1(β) (σ 1 -content), 101 319

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