Summary: We derive an inductive, combinatorial definition of a polynomial-valued regular isotopy invariant of links and tangled graphs. We show that the invariant equals the Reshetikhin-Turaev invariant corresponding to the exceptional simple Lie algebra \(G_ 2\). It is therefore related to \(G_ 2\) in the same way that the HOMFLY polynomial is related to \(A_ n\) and the Kauffman polynomial is related to \(B_ n\), \(C_ n\), and \(D_ n\). We give parallel constructions for the other rank 2 Lie algebras and present some combinatorial conjectures motivated by the new inductive definitions.