2 Th e Knot Book
A knot is just such a knotted loop of string, except that we think of the
string a s havin g n o thickness , it s cross-sectio n bein g a singl e point . Th e
knot is then a closed curve in space that does not intersect itself anywhere .
We will not distinguish between the original closed knotted curv e an d
the deformations o f that curve through spac e that do not allow th e curv e
to pass throug h itself. All of thes e deformed curve s will be considered t o
be th e sam e knot . W e think o f th e kno t a s i f i t wer e mad e o f easil y de -
formable rubber .
Figure 1.2 Deformin g a knot doesn't change it.
In thes e picture s o f knot s (Figur e 1.2) on e sectio n o f th e kno t passe s
under anothe r sectio n at each crossing. The simplest knot of all is just th e
unknotted circle , which w e cal l the unkno t o r th e trivia l knot . Th e nex t
simplest kno t i s calle d a trefoi l knot . (Se e Figur e 1.3.) Bu t ho w d o w e
know thes e ar e actually differen t knots ? Ho w d o w e kno w tha t w e
couldn't untangl e th e trefoi l kno t int o th e unkno t withou t usin g scissor s
and glue, if we played with it long enough?
o
e
a b
Figure 13 (a ) The unknot, (b) A trefoil knot .
Certainly, if you make a trefoil knot out of string and try untangling i t
into the unknot, you will believe very quickly that it can't be done. But we
won't b e able to prove it until we introduce tricoloration o f knots i n Sec-
tion 1.5.
In th e tabl e a t th e bac k o f th e book , ther e ar e numerou s picture s o f
knots. All of these knots are known t o be distinct. If we made an y one of
them ou t o f string , w e woul d no t b e abl e t o defor m i t t o loo k Uk e an y
of th e others . On th e othe r hand , her e i s a pictur e (Figur e 1.4) o f a kno t
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