4-fold symmetric quandle invariants of 3-manifolds

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.