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Knots Unravelled

A guide to the fascinating world of knots, from string to mathematics.

Knots Unravelled: from string to mathematics

Knots Unravelled is a guide to the fascinating world of knots, from the familiar realm of knotted string to the less familiar branch of mathematics known as knot theory.

• Are two given knots the same or different?
• How many knots are there?
• Can knots be classified?

Questions like this are easily asked, but finding answers requires more effort. Mathematical ideas help to put the study of knots on a firm footing, and also either answer such questions, or explain why an answer cannot be found. The core chapters of Knots Unravelled lead the reader through this mathematics, from the basics to the frontiers of current work in knot theory.

Between the main chapters, the ‘interludes' reveal some of the rich variety of ways in which knots appear throughout human culture, drawing attention to related mathematics and making connections with other material in the book.

A key feature of the text is the range of tasks and activities for the reader to work through---with string, rope, or pencil and paper to hand! Complete solutions are provided at the back of the book.

The book makes full use of clear diagrams, and a table of knots, a glossary and an index are included.

Sample pages can be seen in the Book Gallery.

Reviews of Knots Unravelled appear in:

• The European Mathematical Society Book Reviews (January 2012)

• European Mathematical Society Book Review
• Book Gallery
• Order
• Order form (PDF)

“It is a very attractive presentation of elementary knot theory and I shall have no hesitation in recommending it to any beginner wondering how to start thinking about knots.” Raymond Lickorish

Readership

In all but the last chapter of the book the mathematical methods involved are not advanced and should be accessible to any enquiring reader, from around age 12 or 13. The final chapter is more demanding, requiring some familiarity with negative powers and the idea of polynomials. Throughout the book the reader is required to follow careful reasoning and some of the ideas are quite sophisticated.

For the teacher, Knots Unravelled is a useful classroom resource, either for individual study, or as the basis for investigative work.

The authors

Meike Akveld has taught mathematics at school (for 10 years) and university and currently teaches at ETH Zürich, Switzerland. Andrew Jobbings taught mathematics at secondary school in England (for 28 years) before founding Arbelos.

Both authors first came across mathematical knot theory during their studies in topology, and both are keen to bring mathematics to a wider audience. They met in Minsk in 2009 as part of their voluntary work for the Kangourou sans Frontières.

• Meike Akveld
• Andrew Jobbings
• Kangourou sans Frontières

Contents

1. Introduction
   1. Knots everywhere
   2. Knots in rope
   3. Knot science
   4. History

Interlude Knots in paper

2. Working with diagrams
   1. Describing knots
   2. Mathematical knots
   3. Projections and knot diagrams
   4. Knotted or not? The same or different?
   5. Reidemeister moves

Interlude Celtic knots

3. Counting crossings
   1. Telling knots apart
   2. The crossing number
   3. Which crossing numbers are possible?
   4. Does the crossing number classify knots?
   5. Crossing number 5
   6. Classifying knots

Interlude Tie knots

4. New knots from old
   1. Mirror images
   2. Combining knots
   3. Changing crossings

Interlude The figure of eight

5. Using colours
   1. Knot invariants
   2. Three-colourability

Interlude Hunter's bend

6. Links
   1. What is a link?
   2. The Borromean rings
   3. Components
   4. The linking number
   5. Three-colourability

Interlude Torus knots

7. Knot polynomials
   1. The bracket polynomial
   2. The writhe
   3. The X-polynomial
   4. The Jones polynomial

Postlude A special trefoil

Solutions

Bibliography

Table of knots and links

Glossary

Index

List of corrections (PDF)

© A K Jobbings 2004–2012