A History of Mathematics- From Mesopotamia to Modernity

Modernity and itsAnxieties 225

P

a

a

a

e

f

b

b

b

c

de

c

d

e

g

h

i

j

k

k

f

g

i k

g

I

II

II

III

IV

VI V

III

IV

V VI

h

i

k

c

d

QQR

S

O

P

R

O

S

S

O

P

P

Q

R S

O

Q

R

I

Fig. 5The ‘dodecahedral space’ from Seifert and Threlfall’s book. This is obtained from a solid dodecahedron by gluing opposite
faces (marked I, II, III, IV, V) as shown in the picture.

Fig. 6A knot (‘true lover’s knot’) with eight crossings.

The changes which took place can be well illustrated by the specific case of knots. Unlike the
subject-matter of topology in general, these are easy to understand, and what they are, in basic
terms, has remained the same since they were first systematically studied by Tait (in response
to a failed theory of Lord Kelvin) in the 1870s.^10 For that reason, they provide a particularly
interesting index of what does change. One thinks of a knot (Fig. 6) as a closed curve (a ‘circle’)
in three dimensions, and defines two knots to be equivalent if one can be deformed into the other;
so much is more or less obvious. Also immediately clear to Tait, and probably to the reader, is that
one can represent a knot unambiguously by projecting it down into a plane (as in the figure) using
broken lines to mark which strand goes under at each crossing. Of course, this ‘diagram’ is far
from unique, and the very simplest question is how one tells whether two diagrams determine the
same knot. Since that would require a language of diagrams, it is already perhaps too difficult. Tait
determined all knots which had diagrams with up to eight crossings, and made some important and
hard conjectures. This, it must be remembered, waswithoutusing any of the still-to-be invented
tools of topology.

10. Tait referred to earlier work of Gauss and Listing, but he is usually considered the founder.