COMBINATORIAL KNOT THEORY

A classic knot is an embedded simple loop in 3-dimensional space. It can be described as a 4-valent planar graph or network in the horizontal plane, with the vertices or crossings corresponding to double points of a projection. At this stage we have the shadow of the knot defined by the projection. We can reconstruct the knot by lifting the crossings into two points in space, one above the other. This information is preserved at the vertices by cutting the arc which appears to go under the over crossing arc. We can then act on this diagram of the knot using the famous Reidemeister moves to mimic the motion of the knot in space. The result is classic combinatorial knot theory. In recent years, many different types of knot theories have been considered where the information stored at the crossings determines how the Reidemeister moves are used, if at all.

In this book, we look at all these new theories systematically in a way which any third-year undergraduate mathematics student would understand. This book can form the basis of an undergraduate course or as an entry point for a postgraduate studying topology.

Contents:
Roger Fenn: A Life in MathsCombinatorial Knot Theory: An Introduction to Knot DiagramsBasic ConceptsGeneralised Knot TheoriesChord DiagramsVirtual Crossings and Virtual KnotsMax Newman's Proof TechniqueGeneralised BraidsAlexander and Markov TheoremsUnfinished Business from Chapter 7The Sorting Method: InvariantsAppendices:The Homflypt PolynomialCategoriesCoding Knots on SurfacesExamples of Planar DoodlesBibliographyIndex
Readership: Textbook or for self-study for advanced university undergraduate and postgraduate courses on topology. Researchers who wish to extend their knowledge on knot theory.

Roger A Fenn is currently Emeritus Reader at University of Sussex, UK. He has always been interested in maths. He wondered about recurring decimals, extracting square roots and the beautiful Greek symbols in his older brothers' schoolbooks. However, this didn't prevent him from failing the local 11-plus exam which was universal at the time and sent the passers to prestigious establishments such as Bristol Grammar School and the losers to Filton Road Secondary Modern School where, to his surprise, he was in the top stream and began to learn proper maths and even French and German.Filton Road only survived another year and the pupils were moved up the hill to Lockleaze School, the first comprehensive school in Britain. In evidence that if you spend money, you get results. Lockleaze had a large playing field, a physics lab, a woodwork room and a metalwork room with a kiln. But most importantly it had good teachers, a head of music, a head of languages and a head of maths, Colin Evans. It was Colin who taught Roger all he knew and got him into Clare College, Cambridge in 1961, where he was the first 11-plus failure and comprehensive pupil to attend.After Cambridge he did his PhD under John Reeve at King's College, London. There he learnt about geometric topology and proved 'the table theorem' for his thesis which showed you could always place a square table so that it didn't wobble and the top was horizontal. In 1967, he became a lecturer at the new (plate glass) University of Sussex, where he stayed for the rest of his career.According to Research Gate, Roger Fenn has 93 publications with more to come.He is now retired from teaching but still doing maths.He lives in Lewes, Sussex with his wife and enjoys walking, making things and playing music badly.