The paper introduces –fold symmetric quandles and –fold symmetric quandle homotopy invariants of –manifolds. We classify –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 –fold symmetric quandle homotopy invariant. We also discuss symmetric quandle cocycle invariants and coloring polynomials of –fold symmetric quandles.
"$4$–fold symmetric quandle invariants of $3$–manifolds." Algebr. Geom. Topol. 11 (3) 1601 - 1648, 2011. https://doi.org/10.2140/agt.2011.11.1601