Symmetric biquandle cohomology and cocycle invariants for unoriented links and surface-links

We introduce the notion of symmetric biquandles and establish a shadow symmetric biquandle (co)homology theory. This framework provides shadow symmetric biquandle cocycle invariants for unoriented links in three-dimensional space and unoriented surface-links in four-dimensional space, represented by broken surface diagrams. We further interpret these invariants via marked graph diagrams (ch-diagrams) and examine their properties with illustrative computations. In addition, we classify all symmetric non-quandle biquandles together with corresponding good involutions of orders 2, 3, and 4 up to isomorphism.