The definition implies that each defect line
δ
of Σ corresponds either to a single defect line
of Σ
′
with the same label as
δ
, or else splits into several defect lines of Σ
′
that all carry the
same label as
δ
and which are interrupted by fillable gluing circles that are not present in Σ.
Moreover, the gluing boundaries of Σ correspond to identical gluing boundaries of Σ
′
. As an
illustration, the following picture shows the refinement (
D
;
D
′
) of a one-holed disk
D
for which
D
′
has three additional gluing circles:
D
=
D
′
=
(5.26)
Remark 5.13.
(i) For any refinement (Σ; Σ
ref
), by definition every gluing boundary of Σ
ref
that is not a gluing
boundary of Σ is fillable.
(ii) It is worth comparing the notion of refinement with Definition 5.10 of a fillable disk of type
X
: It is easily seen that for every refinement (
X
,
X
ref
) with
X
a defect surface for which
the underlying surface
X
is a disk and which has at most one free boundary segment, the

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- Fall '17