Computation of the knot symmetric quandle and its application to the plat index of surface-links (Q6564527)

A surface-knot is a closed connected surface embedded in \(\mathbb{R}^4\), and a surface-link is a disjoint union of surface-knots. A symmetric quandle is a pair \((X, \rho)\) of a quandle \(X\) and a good involution \(\rho:X \to X\), and one can associate a symmetric quandle to each unoriented surface-link in \(\mathbb{R}^4\). Thus, these algebraic structures are useful for distinguishing unoriented surface-links. \par A surface-link can be presented by the plat closure of a braided surface, called a plat form presentation. In this paper, the author computes the symmetric quandles for surface-links using their plat form presentations. As an application, it is proved that for each pair of integers \(g \ge 0\) and \(m \ge 2\), there exist infinitely many distinct surface-knots of genus \(g\) whose plat indices are \(m\).