Hurwitz equivalence of the degree-three monodromy factorization

Establish the Hurwitz equivalence between the positive factorization of type (4,3) constructed by Baykur and Korkmaz and the monodromy factorization of Xiao’s degree-three genus-two fibration.

Background

Xiao’s genus-two fibration in degree three has four nonseparating and three separating vanishing cycles. Baykur and Korkmaz had previously constructed a positive factorization of the identity in the genus-two mapping class group with the same type, but the relationship between that factorization and the monodromy arising from Xiao’s fibration was not initially known.

The paper subsequently notes that Evan Huang used the author’s computation of the degree-three vanishing cycles to prove the required Hurwitz equivalence. Thus, the passage records an explicitly identified problem that is described as open at the point of its historical formulation, although the surrounding introduction indicates that it was later resolved.