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-