6 Th e Knot Book

Knot theor y i s a subfield o f a n are a o f mathematic s know n a s topol -

ogy. Topology i s the stud y o f th e properties o f geometri c object s tha t ar e

preserved unde r deformations . Jus t a s w e thin k o f th e knot s a s bein g

made of deformable rubber , so we think of the more general geometric ob-

jects in topology as deformable. Fo r instance, a topologist does not distin -

guish a cube from a sphere, since a cube can be deformed int o a sphere by

rounding of f th e eight corners and smoothin g the twelve edges, as in Fig-

ure 1.8.

Figure 1.8 A cube and a sphere are the same in topology.

Topology i s one o f th e majo r area s o f researc h i n mathematic s today .

Work in knot theory has led to many important advances in other areas of

topology. We discuss some of these connections in Chapter 9.

In this book, we investigate the mathematical theory of knots. The em-

phasis is on curren t researc h i n knot theory . Unlike th e situation i n som e

other field s o f mathematics , many o f th e unsolved problem s i n kno t the -

ory are easily stated. Much of the theory is accessible to someone withou t

any background i n upper-level mathematics . There are open problem s i n

the field that can be attacked and perhaps solved by nonexperts.

The best way to learn any kind o f mathematics is by doing mathema -

tics, not just by reading about what others have done. Therefore, through -

out this book there are numerous open problems in knot theory. Try them!

Think t o yourself , "Ho w woul d I solv e thi s problem? " Mayb e yo u ca n

come up with the essential new idea and discover the solution.

Exercise 1.3 Us e string (or an extension cord) to show that the followin g

knot is the unknot.