Hopf algebras and polynomial invariants of combinatorial structures

Abstract 5
Declaration 6
The Author 7
Acknowledgements 8
Introduction 9
1 Preliminaries 13
1.1 G raphs .......................................................................................................... 13
1.2 Graph Polynom ials ................................................................................... 16
1.3 Knots and Links ...................................................................................... 17
1.4 The Medial Construction .......................................................................... 20
2 Graph Polynomials and the Kauffman Bracket Polynomial 24
2.1 Graph Polynom ials ............................................................................... . 24
2.2 The Kauffman Bracket Polynom ial ...................................................... 35
3 Hopf Algebras and Delta Operators 41
3.1 Hopf Algebras .................... 41
3.2 Delta Operators .......................................................................................... 47
4 A Hopf Algebra for Link Diagrams 57
4.1 Colouring Link Diagrams ........................................................ 58
4.2 The Hopf Algebra ..................................................................................... 61
4.3 The Quotient Module ..................................................................... 64
4.4 Delta Operators ........................................................................................ 69
5 A Hopf Algebra for Graphs 75
5.1 The Hopf Algebra ....................................................................... 76
5.2 The Quotient M odule ............................................................................... 80
5.3 The Delta Operator .................................................................................. 85
6 A Hopf Algebra for Signed Graphs and the ^oo-polynomial 90
6.1 The -polynomial for Signed G raphs .............................................. 90
6.2 The Hopf Algebra for Signed G rap h s .................................................. 93
6.3 The Quotient Module ............................................................................... 95
7 A Hopf Algebra for Link Diagrams and a New Link Diagram
Invariant 99
7.1 The $oo-polynomial for Link D iagram s............................................... 99
7.2 The Hopf Algebra of Link Diagrams ......................................................... 101
7.3 The Quotient M odule ................................................................................... 103
7.4 The Medial Construction R evisited ......................................................... 107
8 Hopf Algebra and Polynomial Invariant Relationships 110
8.1 A Hopf Algebra for Link U niverses ......................................................... 110
8.2 The Bracket and 4/oo-Polynomials ............................................................ Ill
8.3 A Hopf Algebra for Planar G ra p h s .................... 112
8.4 A Hopf Algebra for Simple G ra p h s ......................................................... 114
8.5 Relations Between Hopf Algebras of G ra p h s ........................................ 117
A A Graded Hopf Algebra for Link Diagrams 122
B Some Calculations Concerning the ^ -polynom ial for Complete
Graphs 126
Bibliography 131