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
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