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 Maths
-
Combinatorial Knot Theory: An Introduction to Knot Diagrams
-
Basic Concepts
-
Generalised Knot Theories
-
Chord Diagrams
-
Virtual Crossings and Virtual Knots
-
Max Newman's Proof Technique
-
Generalised Braids
-
Alexander and Markov Theorems
-
Unfinished Business from Chapter 7
-
The Sorting Method: Invariants
-
Appendices:
- The Homflypt Polynomial
- Categories
- Coding Knots on Surfaces
- Examples of Planar Doodles
-
Bibliography
-
Index
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.
