  In loop quantum gravity, states of quantum geometry are represented by
classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is
worthwhile to enumerate and distinguish these classes. This paper looks at the
case of 4-regular graphs, which have an interpretation as objects dual to
triangulations of three-dimensional manifolds. Two different polynomial
invariants are developed to characterize these graphs -- one inspired by the
Kauffman bracket relations, and the other based on quandles. How the latter
invariant changes under the Pachner moves acting on the graphs is then studied.
