Sossinsky:Knots. Mathematics with a twist.pdf - English

22 KNOTS
2.6d, 2.6e). Two Seifert circles are nested if one of them is inside the
other and if the orientations of the two circles coincide. Note that
desingularizing a coiled knot always yields a nested system of Seifert
circles, and vice versa (Figure 2.6b).
On the other hand, when Seifert circles are not nested (as in Figure
2.7b), the change-of-infinity' operation nests the two circles (Figure
2.7c). In fact, this figure shows that while circles l and 2 are nested,
circle 3 do es not encompass them; but inverting this last circle results
in the circle 3', which neatly encompasses circles l and 2. (In this case,
the change-of-infinity move resembles the operation carried out in
Figure 2.5.)
Let us now consider the planar map determined by the knot N. A
country in this map is said to be in turmoi[ if it has two edges that
belong to two different Seifert circles, labeled with arrows going in
the same direction around the region. For example, in the smoothing
of knot N in Figure 2.8, the country H is in turmoil, whereas regions
P1 and P2 are not: since the thick edges head in the same direction
around H and belong to two distinct Seifert circles, H is in turmoil; P1
is not, because its edges belong to a single Seifert circle; finally, P2
is not in turmoil either, because its edges go around P2 in opposite
directions.
An operation called perestroika can be applied to any country in
turmoil (Figure 2.9). Perestroika consists in replacing the two faulty
edges by two "tongues:' one of which passes over the other, forming
two new crossings. The result is to create a centraI country (not in turmoil)
and several new countries, some of which (in this case, two)
may be swallowed up by bordering countries. I am sure that now the
reader understands the choice of the geopolitical term perestroika.