Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 233

1bR

1aR

1cL

1aL

1bL

1cR

2bR

2aR

2cL

2aL

2bL

2cR

3bR

3aR

3cL

3aL

3bL

3cR

Fig. 12.13.Bottlenecks

Fig. 12.14.An additional complication

LHT path is affected by how many bottlenecks are present, invalidating
the previous argument.
We resolve this issue in the following way: when a bottleneck is formed,
we count it as half an LHT path. We then only count closed LHT paths that
are formed by gluing together two ends that are not bottlenecks. In this way,
the previous argument providing a logarithmic bound remains intact, at the
expense of raising the constant from 1 to 3/2. This is clearly an overestimate,
since a bottleneck may be destroyed before it is joined to another bottleneck.
This establishes Theorem 7.1.
The argument to establish Theorem 7.2 is similar, but more complicated
(Fig. 12.14). Again the problem is that we have to worry about the creation