  By using the cohomology theory of quandles, quandle cocycle invariants and
shadow quandle cocycle invariants are defined for oriented links and
surface-links via broken surface diagrams. By using symmetric quandles,
symmetric quandle cocycle invariants are also defined for unoriented links and
surface-links via broken surface diagrams. A marked graph diagram is a link
diagram possibly with $4$-valent vertices equipped with markers. S. J.
Lomonaco, Jr. and K. Yoshikawa introduced a method of describing surface-links
by using marked graph diagrams. In this paper, we give interpretations of these
quandle cocycle invariants in terms of marked graph diagrams, and introduce a
method of computing them from marked graph diagrams.

Geometric Topology