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We give an algorithm to decide if a given braid is a product of two factorswhich are conjugates of given powers of standard generators of the braid group.The same problem is solved in a certain class of Garside groups includingArtin-Tits groups of spherical type. The solution is based on the Garsidetheory and, especially, on the theory of cyclic sliding developed by Gebhardtand Gonzalez-Meneses. We show that if a braid is of the required form, then anycycling orbit in its sliding circuit set in the dual Garside structure containsan element for which this fact is immediately seen from the left normal form.

Algorithmic recognition of quasipositive braids of algebraic length two