Neurology Mathematics Ideas71.pdf - The definition implies that each defect line \u03b4 of \u03a3 corresponds either to a single defect line of \u03a3\u2032 with the

Neurology Mathematics Ideas71.pdf - The definition implies...

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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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