Computation of the knot symmetric quandle and its application to the plat index of surface-links

A surface-link is a closed surface embedded in the 4-space, possibly disconnected or non-orientable. Every surface-link can be presented by the plat closure of a braided surface, which we call a plat form presentation. The knot symmetric quandle of a surface-link F is a pair of a quandle and a good involution determined from F. In this paper, we compute the knot symmetric quandle for surface-links using a plat form presentation. As an application, we show that for any integers g >= 0 and m >= 2, there exist infinitely many distinct surface-knots of genus g whose plat indices are m.