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