10 Th e Knot Book
into a unique set of prime numbers, a composite knot factors into a unique
set of prime knots.
The appendix table, which contains projections of knots, and is located
at th e bac k o f th e book , list s onl y th e prim e knot s an d doe s no t includ e
any composit e knots . It's like a table of prime numbers . Although al l the
positive integers aren't listed, any integer can be constructed by taking the
appropriate product of the primes that are listed.
Exercise 1.8 Usin g th e appendi x table , identif y th e facto r knot s tha t
make up the composite knot in Figure 1.15.
Figure 1.15 A composite knot.
Exercise 1.9 Sho w that the knot in Figure 1.1 6 is composite.
Figure 1.16 Anothe r composite knot.
One way tha t compositio n o f knots does differ fro m multiplicatio n o f
integers is that there is more than one way to take the composition o f tw o
knots. We have a choic e of where w e remov e th e ar c from th e outside of
each projection . Wil l this choic e affect th e outcome ? Surprisingly , th e an -
swer is yes. It is often possibl e to construct tw o different composit e knot s
from the same pair of knots / and K.
We firs t nee d t o pu t a n orientatio n o n ou r knots . A n orientatio n i s
defined b y choosing a direction to travel around th e knot. This direction is
denoted by placing coherently directed arrow s along the projection o f th e
knot in the direction of our choice. We then say that the knot is oriented.
When w e the n for m th e compositio n o f tw o oriente d knot s / an d K,
there are two possibilities. Either the orientation o n / matche s the orienta-