We give fundamental moves for the neighborhood equivalence classes of spatial trivalent graphs. We define a coloring invariant and a cocycle invariant for the neighborhood equivalence classes and then for all spatial graphs. We show that the cocycle invariant detects the chirality of a knotted handlebody.
"Moves and invariants for knotted handlebodies." Algebr. Geom. Topol. 8 (3) 1403 - 1418, 2008. https://doi.org/10.2140/agt.2008.8.1403