Topology in Molecular Biology

222 R. Brooks

Fig. 12.10.Various options for orientations on the cube

got surfaces of genus two, because there are two LHT paths (two of length 12
in the third example, one of length 20 and one of length 4 in the fourth).
It is difficult to decide which surfaces are represented by the last two
orientations, mostly because it is difficult to find names for surfaces of genus
two.

12.6 The Ahlfors–Schwarz Lemma

The Ahlfors–Schwarz lemma should be familiar to students of complex analy-
sis. Denoting byDthe unit disk

D={z∈C:|z|< 1 },

it states

Lemma 1 (Schwarz).Letf:D→Dbe a holomorphic map, which takes 0
to 0.
Then
|f′(z)|≤ 1

and
|f(z)|≤|z|,z=0,

with equality in either of the two inequalities at any point if and only if

f(z)=eiθ

for someθ.