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We first construct a genus zero positive allowable Lefschetz fibration overthe disk (a genus zero PALF for short) on the Akbulut cork and describe themonodromy as a positive factorization in the mapping class group of a surfaceof genus zero with five boundary components. We then construct genus zero PALFson infinitely many exotic pairs of compact Stein surfaces such that one is acork twist of the other along an Akbulut cork. The difference of smoothstructures on each of exotic pairs of compact Stein surface is interpreted asthe difference of the corresponding positive factorizations in the mappingclass group of a common surface of genus zero.

A genus zero Lefschetz fibration on the Akbulut cork