An Algebraic Approach to Symmetry with Applications to Knot Theory

List of Figures......Page 6
Quandles......Page 8
Involutory quandles......Page 10
The algebraic theory of conjugation......Page 12
Representation of quandles as conjugacy classes......Page 13
Representation of quandles as cosets......Page 14
Algebraic connectivity......Page 15
n-Cores......Page 16
Examples of simple quandles......Page 17
Classification of simple p-quandles......Page 19
Augmented quandles......Page 21
Quotients of augmented quandles described by normal subgroups of the augmentation group......Page 25
Involutory quandles and geodesics......Page 28
Involutory quandles generated by two points......Page 29
Group cores......Page 30
Distributive quandles......Page 32
Moufang loop cores......Page 35
Distributive 2-quandles with midpoints......Page 38
The fundamental quandle of a pair of spaces.......Page 40
The fundamental quandle of a disk......Page 44
The Seifert-Van Kampen theorem......Page 45
Applications of the Seifert-Van Kampen theorem......Page 47
Knot quandles......Page 48
A presentation of the knot quandle......Page 49
The invariance of the knot quandle......Page 51
A presentation of the knot quandle (continued)......Page 52
A representation of the knot quandle......Page 54
The Alexander invariant of a knot......Page 55
The involutory knot quandle......Page 57
Bibliography......Page 60
Index......Page 61