12.3 Plumbing Fixture
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also introduce a minimal length s 0 > 0, called the stub parameter, for the tube. In
this case, the plumbing fixture parameter is generalized to
q = e
−s+iθ ,
s ∈ [s 0 , ∞),
θ ∈ [0, 2π),
s 0 ≥ 0.
(12.53)
What is the effect on the subspaces F g,n (s 0 ) and V g,n (s 0 )? Obviously, less surfaces
can be described by the plumbing fixture if s 0 > 0 than if s 0 = 0, since the plumbing
fixture cannot describe anymore surfaces that contain a tube of length less than s 0 .
Equivalently, the values of the moduli described by the plumbing fixture is more
restricted when s 0 > 0. More generally, one has
s 0 < s
0 :
F g,n (s
0 ) ⊂ F g,n (s 0 )
V g,n (s 0 ) ⊂ V g,n (s
0 ).
(12.54)
This is illustrated in Fig. 12.10. Even if s 0 is very large, V g,n still does not include
surfaces arbitrarily close to degeneracy. In general, we omit the dependence in s 0
except when it is necessary.
Fig. 12.10 In light grey is the subspace covered by the V 0,4 (s 0 ) as in Fig. 12.9. In dark grey is the
difference δV 0,4 = V 0,4 (s 0 + δs 0 ) − V 0,4 (s 0 ) with δs 0 > 0