Introduction 11
tion on K in J#K, resulting in an orientation for J#K, or the orientation on /
and K do no t matc h u p i n J#K. Al l of th e composition s o f th e tw o knot s
where th e orientation s d o matc h u p wil l yiel d th e sam e composit e knot .
All o f th e composition s o f th e tw o knot s wher e th e orientation s d o no t
match u p wil l als o yield a singl e composit e knot ; however, i t is possibl y
distinct fro m th e composit e kno t generate d whe n th e orientation s d o
match up (Figur e 1.17).
KS
®W %p
a b c
Figure 1.17 (a ) Orientation s match , (b ) Orientation s match , (c ) Orienta -
tions differ .
To convince ourselves that the first two compositions in Figure 1.1 7 re-
ally d o giv e u s th e sam e knot , w e ca n shrin k / dow n i n th e firs t pictur e
and then slide it around K until we obtain the second picture (Figure 1.18).
Although thi s wil l not b e th e cas e in general , in thi s particula r example ,
the thir d compositio n i n Figur e 1.1 7 als o gives th e sam e kno t a s th e tw o
preceding compositions . This occurs because one of the factor knot s is in-
vertible. A knot is invertible if it can be deformed bac k to itself s o that a n
orientation o n it is sent to the opposite orientation. In the case that one of
the tw o knot s i s invertible , sa y / , w e ca n alway s defor m th e composit e
knot so that the orientation on K is reversed, and hence so that the orienta-
tions of / and K always match. Therefore, there is only one composite kno t
that we can construct from th e two knots.
Figure 1.18 Tw o compositions that are the same.
The first knot that is not invertible in the table at the end of the book is
the knot 817. Composing i t with itsel f i n the two differen t way s produce s
two distinct composite knots that are not equivalent (Figur e 1.19). In order
Previous Page Next Page