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

Abstract

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 [Formula: see text] is a pair of a quandle and a good involution determined from [Formula: see text]. 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 [Formula: see text] and [Formula: see text], there exist infinitely many distinct surface-knots of genus [Formula: see text] whose plat indices are [Formula: see text].