  It is known that every surface-link can be presented by a marked graph
diagram, and such a diagram presentation is unique up to moves called Yoshikawa
moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum
A_2 invariant, for tangled trivalent graph diagrams. In this paper, a
polynomial for a marked graph diagram is defined by use of the quantum A_2
invariant and it is studied how the polynomial changes under Yoshikawa moves.
The notion of a ribbon marked graph is introduced to show that this polynomial
is useful for an invariant of a ribbon 2-knot.
