that if you want to make a map, you only need to use four color s to make
sure tha t n o tw o countrie s o f th e sam e colo r touc h eac h othe r alon g a n
edge. This was the first proof o f a major theore m that used computer s ex-
tensively t o enumerate th e thousands o f case s that nee d t o be examined .
(See Appel and Haken, 1977).
Why should anyone be interested in knots? What's so important abou t
being abl e t o tel l whether a tangled-up loo p o f strin g i s truly tangle d o r
can in fact be untangled without cutting and gluing ?
Much of the early interest in knot theory was motivated by chemistry .
In the 1880s, it was believed tha t a substance called ethe r pervade d al l of
space. In an attemp t t o explain th e differen t type s o f matter, Lord Kelvi n
(William Thomson , 1824-1907 ) hypothesize d tha t atom s wer e merel y
knots in the fabric o f this ether. Different knot s would the n correspond t o
different element s (Figure 1.7).
This convince d th e Scottis h physicis t Pete r Guthri e Tai t (1831-1901)
that if he could list all of the possible knots, he would be creating a table of
the elements. He spent many years tabulating knots. At the same time, an
American mathematician named C. N. Little was working on his own tab-
ulations for knots.
He? Pb ? Ni ?
Figure 1.7 Atom s are knotted vortices?
Unfortunately, Kelvi n wa s wrong . I n 1887, the Michelson-Morle y ex -
periment demonstrate d tha t ther e was n o ethe r t o knot. A more accurat e
model o f atomi c structur e appeare d a t th e en d o f th e nineteent h centur y
and chemists lost interest in knots for the next 100 years. But in the mean-
time, mathematician s ha d becom e intrigue d wit h knots . A centur y o f
work on the mathematical theory of knots followed .
Interestingly enough, in the 1980s, biochemists discovere d knottin g in
DNA molecules. Concurrently, synthetic chemists realized it might be pos-
sible to create knotted molecules , where th e type o f kno t determine d th e
properties o f th e molecule . A mathematica l fiel d tha t wa s bor n ou t o f
a misguide d mode l fo r atom s ha s turne d ou t t o hav e severa l significan t
applications t o chemistr y an d biology . W e discus s thes e application s i n