Introduction 3

that i s actuall y a trefoi l knot , eve n thoug h i t look s completel y differen t

from the previous picture of a trefoil.

Figure 1.4 A nonstandar d picture of the trefoil knot .

Exercise 1.1 Mak e thi s knot ou t o f strin g an d the n rearrang e i t to sho w

that i t is the trefoil knot . (Actually , an electrica l extension cor d work s

better than string. You can tie a knot in it and then plug it into itself i n

order t o form a knot. A third optio n is to draw a sequence of picture s

that describ e th e deformatio n o f th e knot. This is particularly eas y t o

do o n a blackboard, wit h chal k an d eraser . A s yo u defor m th e knot ,

you ca n simply eras e and redra w th e appropriat e section s o f th e pic-

ture.)

There are many differen t picture s o f th e sam e knot . In Figure 1.5, w e

see three different picture s of a new knot, called the figure-eight knot. We

call such a picture of a knot a projection of the knot.

(P®( p

Figure 1.5 Thre e projections of the figure-eight knot .

The place s wher e th e kno t crosse s itsel f i n th e pictur e ar e calle d th e

crossings o f th e projection . W e sa y tha t th e figure-eigh t kno t i s a four -

crossing kno t becaus e ther e i s a projectio n o f i t wit h fou r crossings , an d

there are no projections of it with fewer than four crossings .

If a kno t i s t o b e nontrivial , the n i t ha d bette r hav e mor e tha n on e

crossing in a projection. Fo r if it only has one crossing, then the four end s

of the single crossing must be hooked u p i n pairs in one of the four way s