Index of Notation
N (nonnegative integers)
Z (integers)
Q (rationals)
R (reals)
C (complex numbers)
Introduction
Bn (braid group), 1
Bn
+
(braid monoid), 1
Bn∗ + (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
er
min
(Φ3-normal form), 40
Chapter III
RG (group algebra), 44
L2(G) (Hilbert space), 45
b
β (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
Qsh(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
Fn LDM (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) (σ1-content), 101
319
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