Introduction 7

Exercise 1.4 Sho w tha t an y kno t ha s a projectio n wit h ove r 1000 cross-

ings.

Certain types of knots are particularly interesting. One such type is an

alternating knot . An alternatin g knot is a knot wit h a projection tha t ha s

crossings that alternate between over and under a s one travels around th e

knot in a fixed direction . The trefoil kno t in Figure 1.3 is alternating. So is

the figure-eight kno t in Figure 1.5, since the two projections of it on the left

and middle are alternating.

Exercise 1.5 Choos e crossing s a t eac h verte x i n Figur e 1.9 t o mak e th e

resulting knot alternating.

Figure 1.9 A projectio n without over- and undercrossings .

Exercise 1.6* Show that by changing the crossings from ove r to under o r

vice versa, any projection o f a knot can be made into the projection of

an alternatin g knot . (Thi s isn't a s easy as it might seem . How d o yo u

know your procedure will always work?) In a projection wit h n cross-

ings, what is the maximum number of crossings that would have to be

changed in order to make the knot alternating ?

Exercise 1.7* Show that by changin g som e of the crossings from ove r t o

under or vice versa, any projection of a knot can be made into a projec-

tion of the unknot.

1.2 Compositio n of Knots

Given two projections o f knots, we can define a new kno t obtaine d b y re-

moving a small arc from eac h knot projection an d then connecting the fou r

endpoints by two new arcs as in Figure 1.10 . We call the resulting knot th e

composition of the two knots. If we denote the two knots by the symbols /

and K, the n thei r compositio n i s denote d b y J#K. We assum e tha t th e

* Exercise with asterisk denotes more difficult problem .