$4$–fold symmetric quandle invariants of $3$–manifolds

Abstract

The paper introduces $4$–fold symmetric quandles and $4$–fold symmetric quandle homotopy invariants of $3$–manifolds. We classify $4$–fold symmetric quandles and investigate their properties. When the quandle is finite, we explicitly determine a presentation of its inner automorphism group. We calculate the container of the $4$–fold symmetric quandle homotopy invariant. We also discuss symmetric quandle cocycle invariants and coloring polynomials of $4$–fold symmetric quandles.