...Algebraic and combinatorial techniques of knot theory based on arc colouring By Ayodele Arubi Advisor: Dr. Alexei Vernitski A project submitted in partial fulfilment of the requirements for the Degree of Bachelor of Science with Honours in Mathematics University of Essex Colchester, Essex April 2015 Contents Abstract 4 Dedication 4 Acknowledgments 4 Introduction 5 The history of Knot theory 5 Brief history of knot theory 5 Development of the Knot Theory in Physics 7 Counting knots 9 The modern knot theory 12 Defining a Knot 14 Wild Knot 15 Definition of a Knot 16 Differentiating Knots 18 Orientation 18 Knot arithmetic 19 Modular arithmetic 20 Equivalence relation 21 Additivity Property 22 Multplicitivy Property 23 Knot Invariants 24 Reidmeister moves 24 Fox colouring 26 Dehn Colouring 28 Alexander-Briggs colourings 30 Implications of knot colourings 31 Other Knot invariants 31 The linking number 31 Quandles and Racks 35 Brief history about the Rack and Quandle invariant 35 Kei 36 Quandle 37 Racks 38 Definition 38 Examples of racks 39 Important definition of Racks 42 The free product. 42 The Cartesian product. 42 The disjoint union 42 Orbits and stabilizers of racks. 43 The operator group 43 The associated group 43 Bibliography 46 Abstract This project narratively examines the history of the knot theory, its invariants and the theory of arc-colouring. This project will begin...
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