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what are definitions of born or die (birth-death point) and crossing point?

in this paper we have : A presentation for the mapping class group of a closed orientable surface.by Hatcher.W.Thurston ...(a) $f_{t_{0}}$ has exactly one degenerate critical point, of the form $f_{t} …
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configurations of three saddles on one level [duplicate]

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have : There are sixteen possibl …
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