Introduction 9

any two nontrivial knots, we call it a prime knot. Both the trefoil knot an d

the figure-eight kno t are prime knots, although this is not obvious.

For the knot J#K in Figure 1.10 , it is clearly composite. We constructed

it to be. But how abou t the knot in Figure 1.13 ? I s it composite? I n fact, i t

is. I f yo u mak e i t ou t o f strin g an d pla y aroun d wit h th e knot , yo u ca n

eventually get it into a projection that shows that it is composite.

Figure 1.13 A potentially composite knot.

Here's a strange r question . I s the unkno t composite ? Obviously , fro m

the picture i n Figur e 1.3a, i t doesn' t loo k composite . But maybe ther e i s a

way to tangle the unknot u p s o that we get a projection o f it that makes it

obviously a composite knot. That is, perhaps there is a picture of the unknot

that has a nontrivial knot on the left, a nontrivial knot on the right, and tw o

strands of the knot joining them (Figure 1.14). Maybe that part of the projec-

tion corresponding to the knot on the right somehow untangles that part of

the projection corresponding to the knot on the left, resulting in the unknot.

Figure 1.14 Coul d this untangle to be the unknot?

It's somewha t disconcertin g t o realize that i f the unknot wer e a com -

posite knot, then every knot would b e a composite knot. Since every kno t

is the composition of itself with the unknot, every knot would be the com-

position of itself with the nontrivial factor knot s that made up the unknot .

In fact, much to our relief, the unknot is not a composite knot. There is

no wa y t o tak e th e compositio n o f tw o nontrivia l knot s an d ge t th e un -

knot. We use surfaces t o show thi s in Section 4.3. We can think o f thi s re-

sult a s analogou s t o th e fac t tha t th e intege r 1 is not th e produc t o f tw o

positive integers, each greater tha n 1. Moreover, just a s an integer factor s