8 Th e Knot Book

/ K J#K

Figure 1.10 Th e composition J#K of two knots / and K.

two projection s d o no t overlap , an d w e choos e th e tw o arc s tha t w e re -

move t o be on th e outside o f eac h projection an d t o avoid an y crossings .

We choose the two new arc s so they d o not cros s either th e original kno t

projections or each other (Figure 1.11).

New unwanted crossin g

^ New unwanted crossin g

Figure 1.11 Not the composition of / and K.

We call a knot a composite knot if it can be expressed a s the composi-

tion of two knots, neither of which is the trivial knot. This is in analogy to

the positive integers, where we call an integer composit e if it is the prod -

uct of positive integers, neither of which is equal to 1. The knots that make

up the composite knot are called factor knots.

Notice that if we take the composition of a knot K with the unknot, the

result i s agai n K, just a s whe n w e multipl y a n intege r b y 1, we ge t th e

same integer bac k agai n (Figur e 1.12) . If a knot i s not th e compositio n o f

K unkno t K

Figure 1.12 K#(unknot ) is just K.